Guard the places where new Draugr will appear
If there is one location where the new Draugr will regularly appear you will want to have some guards trained in fast dispatching of Draugr to be stationed nearby. There should be multiple people and they should be far enough away to not be overwhelmed by half a dozen or so emerging at the same time, but I take it that they are not going out there en masse as a big army or your problem wouldn't be a continuous slowly rising threat.
The same would be done with people who drown. If you know someone drowned in the river and you can't reach them without having to fear drowning yourself you would station people nearby to deal with maybe a handful off Draugr. Better safe than sorry, you don't want a single guard at the river and suddenly dealing with more Draugr than necessary.
Other than that all dead would be burnt as a general rule of thumb. Again, better safe than sorry. Who knows, maybe they can get the recently deceased out of their grave after some time and put them in the ocean to revive them? People will have a lot of additional superstitions and prefer everyone to be burnt as a general precaution.
As they are not the brightest and not the strongest they pose a little threat and there is not really any need to go against them on a large scale. Slowly decreasing their number will put them back into a reasonable number range sooner or later. If you pay the guards enough this will be a great job opportunity - guards that have to be trained and paid, outposts that have to be built, rations and weapons that have to be delivered, ... It's probably good for your economy.
If their numbers are so high that they will pose a more serious threat you should think about securing common river passages better. Again, guards are a great thing to make sure that you at least know where someone died and will be coming back to life soon. Just don't take money for people to cross the bridges or they will find other ways and then you are back to the root problem.
If there are a lot of these things you should make sure that no small villages will be overrun. The biggest problem with zombie-like hordes is if they start with small villages and are suddenly able to take big villages - because then large cities are not far anymore... Your higher ranked people should therefore reinforce the small villages, which will probably again be good for commerce.
Basically you have the most harmless zombie apocalypse scenario I have ever encountered on this site - good job and have fun with your undead enhanced economy!
If you want them a lot more dangerous than you have proposed - Only allow certain people near oceans/rivers and make drowning someone a serious crime
First problem: undead rot. If you have rotting undead they don't have a lot of time to make more rotting undead, so they won't be able to formulate a bigger strategy. Especially because "kill people, make more Draugr" is currently their only motto. Change this to "Kill all the people and make them ever-lasting Draugr" to get a far more serious threat.
Next problem: communication. You need to have a way for them to talk to each other. If you can give them telepathy of some kind to communicate with other Draugr via magic you would get perfect communication that can't be intercepted and can be used to change battle strategies extremely fast. Without that they are basically humans, so in theory written communication would still be possible. But paper and water don't go too well with one another and leaving message written in the sand of your beaches will be akin to crop circles - you can't get much more obvious.
Problem number three: Intelligence. You need to raise their intelligence, cunning and patience to make them a more capable threat. A bunch of children that drowned in the nearby river will hardly be a threat for a castle. A bunch of enhanced war strategists that were fighting a naval battle might be better, but naval battle is still different from battles on the land and if they drown they probably weren't that good or they would have killed the other one or would have seen the storm coming. Of course there can be any number of additional reasons, but capable strategists are often not fighting on the frontlines so the dead from the frontlines won't be the best strategists that your capital can send to fight this threat.
Four: Make them stronger and faster. You want them to drag people back into the ocean/river/... That requires strength. And it takes a long time during which you are incredibly vulnerable. To take a former example: a bunch of kids that drowned in the local river will hardly be able to drag the village's smith from the village center to the river. No additional Draugr and any searching party, no matter how inexperienced, will find their trail and pick them off while they are a basically non-moving target trying to drag something that is far too heavy for them. This gets easier with adult Draugr, but only so much - have you ever tried dragging a body across the ground for a few kilometers alone? (I hope not, but this should be very, very difficult for your Joe Average. So give your Joe Draugr more power so he's back to hunting faster.)
Five: Your Draugr need to arrange raiding parties. But raising an army takes time and you need to communicate across distances and you need spies. If you have your Draugr be faster and more patient you could have Draugr spies, but currently any spy would easily be detected. Still, fishing villages would notice real quick that there is an army of undead in their waters. And if you kill every fisher that comes too close the village will know that there something is not right. This happens two or three times, they send someone to the capital, the capital sends someone to deal with the Draugr. A little war party should be enough - the Draugr have at most the drowned from the latest naval battle and a village of untrained Draugr at their disposal.
Six: Drowning someone to murder them becomes a serious crime and is something you have to take care of because you suddenly have Draugr behind your walls. Imagine it works by simply drowning someone in a bucket of water - your Draugr, if he is really intelligent, could kill quite a few people without being detected and suddenly you have an attack from the insight. Combine this with telepathy and you can get some reasonable attacks - though you would need a human to be extremely careless or extremely hostile towards the city inside the walls for this to work. To counter this drowning someone would be a serious crime that will be punished by torture of you, your family, your pets, everyone you care about - you could unleash a zombie hell after all. Together with the supserstitions you would therefore have more laws to handle deaths and crimes in that direction.
If you want to make them a threat
Have all animals be susceptible to the Draugr curse - and all Draugr be capable of communication and comparable human-like intelligence. If it's not limited to humans and allows all Draugr, no matter the former species to communicate, things start to get interesting. People wonder why the hunting grounds are less dense when suddenly an army of wolves and boars rise from the tides to swallow whole villages in a sweeping attack with the humanoids carrying the prey off to make more infantry soldiers. It would still be a somewhat known threat as you say they have been around before, so people would know the signs and some may be prepared. But some would not be prepared and suddenly you have localized raids that start to grow into a mass scale attack. This would be horrifying zombie apocalypse - but it would need a lot of time and patience on the side of the Draugr where they are trying to covertly infiltrate the human regions and amass more soldiers for their army. This means a lot of time for people to figure out how to fight them and to prepare counter-attacks. If they are not vastly stronger it would be a normal battle and the best commander on each side might play the most important role in deciding humanities faith.