I have not read all the posts in answer to your OP, but I thought I would share a few thoughts with you. I spent most of my career in management, including buying and "integrating" newly acquired companies into our corporation. In almost all cases it was a one to two year period of disappointment for the newly acquired company and most of the employees, but not all.
First, I gather you are not going to leave the area in search of employment if things don't work out with the new guys. If that is the case you have nothing to lose by easing into your decision. Gather information. Don't be pressured into signing the new agreement till you are ready. If they press you about it just explain that you need more time. Spend the next week or two gathering information. Try to find out where they have other hospitals. Drive over there, have lunch in their cafeteria and meet some of the employees. Get their employee handbook if you don't already have it. Ask question of their employees about their policies. One question to ask is how strict are they to the policy, do they make exceptions. Some companies do, some don't.
Talk to the HR person who made you the offer and ask if they understand what you do. If they don't know and don't try to understand or don't seem to care, that is not a good sign. In many instances the acquiring company sends weaker managers to the new company and they won't have a clue how to talk to you honestly. If they are stronger or senior management that knows anything they will want to understand. Once they do ask them how they will structure your job. When my old company acquired a business I would go in and interview every senior manager to make an assessment. When I found valuable employees that ran the place, regardless of their title I would spend time with them. I always made it my business to interview skilled or technical people who kept the place running.
If you don't get any warm and fuzzies don't give up.
You have already received a lot of good advice. If you don't see a labor attorney at least find a friend that is in HR that can brief you on labor law in your state. After you have done your diligence you can at least start to make a decision. The biggest problem I see from your description of your job is that there is a good chance they won't understand your value. If they do they may not have the flexibility to put you in the appropriate position (weak management). In a lot of smaller organization people like you are what makes the company profitable but only the old management team knows that. (If you have allies in the old team sound them out for advice.)
Finally, the whole thing about salary vs. hourly; other than the numbers, which sort themselves out there is not much to worry about. I spent my whole career in a shirt and tie only to find out when I retired that I liked working with my hands much better. Now, I did not make a mistake being it management, it was very rewarding, but it sure came as a surprise to me how much I enjoyed the hourly side once I got there.
I suspect the biggest decision you have to make is "am I going to like this place once they take over?" There is a real risk you won't. But before you decide ask yourself this question. Do I not like it because I don't like it or do I not like it because it's different than what I am used to? If you don't like it leave. But if it's different you may have to give it a year or two to really know. I have seen many people make the mistake of leaving because it was different, and go to a place that was worse and yet they would accept almost anything from the new company. If your job is as unique as it sounds you will really have to make an adjustment regardless what you decide. Good luck to you