Laws vary from state to state. Also, timeshare interests vary from timeshare to timeshare. If you interest is deeded as most are in the US, your contract will need to meet the requirements for a contract to sell real estate in the state where the timeshare is located. In most, if not all, states, there are standard forms for this approved by that states Bar Association and / or its Board of Realtors. Using one of those from the state where the timeshare is located should help in getting a valid contract. Of course, people can still sometimes screw up on how they fill it out.
Some states, like North Carolina, require that contracts and deeds for the sale of an interest in real property be prepared by attorneys licensed in that state. Reputable multi-state timeshare closing companies retain local counsel where the law requires it. The firm mentioned above is run by an attorney, so I strongly suspect, they would make sure they comply with such local requirements but I am not familiar enough with them to absolutely state that as a fact. To be certain, it would not hurt to ask that question.
Some of the timeshare closing firms run by non-attorneys have had a tendency to play fast and loose with legal requirements, but I would have a higher degree of confidence in a closing firm run by an attorney.
Sometimes, checking with the office of the timeshare involved may find you a good source of help. Some HOA's have their own attorneys who will help with closing at a discount price.