This question has been answered here:
In the movie The BFG giants are carried by helicopter. The answer on movies stack concludes existing transport helicopters could do the job. I think the trolls in the question may be slightly larger than the movie giants are calculated to be, but 2 helicopters can team up to lift cargo suspended beneath in a net, as the giants are. I found images of helicopters carrying a jumbo jet in this manner.
As re sedation I suggest ketamine.
Ketamine is considered a dissociative anesthetic. This means that the drug distorts the
users perception of sight and sound and produces feelings of
detachment from the environment and ones self. The drug also has
anesthetic properties that have been used in both human and veterinary
medicine, and is currently a Schedule III controlled substance. For
humans it has been used in radiation and burn therapy, treatment of
battlefield injuries, and for children who have adverse reactions to
Ketamine is safe as regards cardiopulmonary toxicity and it is also an excellent veterinary drug if the trolls are closer to animals. They will be awake, breathing, but just chilling out in the net. They will be disinclined to struggle and hurt themselves.
Also, 10,000 is a lot of trolls. Triage will be necessary. To save the species, you will want to prioritize juveniles - easier to move, possibly less aggressive and with a longer reproductive life. Adult rescue should prioritize mothers of dependent juveniles.
Large trolls could be moved via drone boat powered floats. Here is a float made of 4 blocks of polystyrene foam adequate to move a 31,000 lb dam part.
You would need more blocks to float a troll but that is fine. Floats can be cheaply assembled with off the shelf parts and attached to drone boats. A satellite uplink will allow random available boats to be converted to drone boats. They will moor off shore. Mockup dummy trolls can be positioned aboard some of the floats so trolls on land will see that trolls can ride these things. When the volcano erupts, trolls will move towards safety. They cannot swim but they are tall and they can wade and the floats are within wading distance. Even if they have not waded, the dummy trolls will hopefully inspire them. They will wade out to the floats and climb on. Once there is a troll passenger the drone boat will head out to sea with troll in tow.
At sea, trolls who jump around will fall off and drown. Each vessel has only 1 troll passenger at risk. Vessel / float failure means that passenger drowns. Calm and lucky trolls will make the ride in peace.
If the trolls are too stupid to figure that out, then those remaining after the airlift will not be rescued. If they wade out and destroy the float, too bad for them. If the volcano erupts with such violence there is no time to seek safety, the leftover trolls die and you might lose your floats and drone boats. No humans will be at risk. Cheap boats and cheap floats can be used: low risk, high yield.