Who should be responsible for those tasks?
That is easy: Lawyers, volunteers trained in writing law, but chosen by lottery to be hired for the purpose on a law-by-law basis. You do not want writing law to be their career, they are not politicians.
Your bigger problem is how to propose laws in this society. You need a way for regular citizens to actually propose laws and get them voted upon without overwhelming the populace with votes on whether dog owners must provide effective doggie shoes when walking their pets in temperatures below 29F, or whether it is legal to have a sexy Santa and sexy female elves in your yard at Christmas.
Or you could have an elected Congress that proposes and writes laws, but all they do is put them up to a general vote. That is less than ideal, because they can still be corrupt and many laws that the majority of people would vote for will never be proposed, due to corruption.
By request, an example of referendum. This is an example only, a full description would require pages of rules, and I don't want to enter into a lengthy discussion of details or why the rules I present are necessary.
Here is one form of referendum: Have a website vaguely like a merger of Stack Exchange [SE] and Kickstarter [KS]. Ideas for laws, in categories like SE, without legalese, can be presented.
People can (like SE) point out that they are duplicates or off topic, etc. You can only sign up with a valid voter ID, but then you can vote ONCE on proposed ideas. Up, down, or Duplicate. You can comment; set rules to stop trolls, too.
At some thresholds, e.g. after 90 days with at least 5000 votes and at least 67% in favor, taxpayer money hires lawyers (see my ans) to draft a law, in full legalese, within the constitutional rights. They will also state, in plain language, how each element written corresponds to a requirement of the law, or was necessary to close a loophole or address an ambiguity in the proposed idea. Or they can state (again by 67% majority of them) that it is an unworkable idea, would violate a previous law that would have to be repealed first, or is impossible to write as law within the Constitution. In that case, the law gets ONE more chance with a new group of randomly chosen lawyers, if they also agree it is unworkable, it is dead.
If it is workable the resultant text is posted. Other lawyers and citizens can comment on it. In the end, this is revised until at least 50% of the original voters vote to ratify the referendum. Alternatively, 50% could vote to fire one or more of the lawyers and have them replaced (by lottery again). I'd only allow that once, however.
Once the referendum is ratified, it goes to the general populace. All the website commentary, the lawyerly discussion, and the original idea remain available, permanently, though the proposal is closed and no more discussion is permitted.
Then the people vote: If 67% vote in favor, the referendum becomes law. Otherwise it fails. Either outcome is recorded on the website for that proposal; and if basically the same thing is proposed in the future, it can be flagged as a duplicate. I'd let voters sign up to be notified if another proposal like it is flagged as a duplicate; they may wish to vote on it again within the 90 day window. (e.g. if they felt it was a hateful law).
I'd make it a mandatory five year prison term crime to tamper with the website, hack it, write code that biases voting, conduct a denial of service attack on it, spoof it, or for any service providers to provide lesser service for it, or charge anything for access to it. Access to this one website is a mandatory public service. I'd provide it in five major languages.
As to why 67%: It is a stable vote. It takes only a few votes to go from 51/49 to 49/51 on a law, and the populace is changing, and you cannot count on all interested parties to vote. However, to go from 67/33 to 33/67 requires more than half of the YES votes to change their mind to NO votes, and that is far less likely to happen if you have thousands of votes. For the entire populace, it may take several decades for such a change to happen, making the law far more likely to reflect the will of most of the people. That is what I mean by "stable", that although the exact percentage of people in favor may vary, the fact that a majority remain in favor will not change very rapidly at all. I think that is important for any law.
I would say legal modification or repeal should follow the same formula: 67% must be in favor of the modification or repeal.
And finally, I would still make it the duty of lawyers to refuse to compromise fundamental constitutional rights that may exist: Like freedom of speech, religion, right to a trial by jury, etc. Unlike the USA, I would demand a much higher threshold to repeal any fundamental rights, perhaps even 90% in favor of repeal. It would also be their job to refuse to pass any law that conflicts or would restrict an existing right or require violation of an existing law or reduce an existing protection.
Of course your story is your story, this is just an example to show such a thing is possible. Rich people can run ads all they want, they cannot stop the commentary or prevent people from voting in the first 90 days. Make it illegal to offer either money or gifts for any vote on the website, with the same mandatory five year prison sentence for both the briber and recipient.