You should definitely check the oil level first, that will take 10 seconds to do. If the oil level is full and the oil is relatively clean then odds are in your favor that the engine is good and the drive is bad. If the oil level shows empty or if the oil is mixed with water, then obviously that would not be good.
1. Should I assume pulling the sterndrive is as expensive as pulling the engine?
No. Pulling a sterndrive is 20 minutes worth of work, pulling the engine can take anywhere from 4 to 6 hours.
2. If it's either one, what are the solutions; a whole new sterndrive or engine or can be repairs be done to lessen my expense?
Sterndrives can be rebuilt. When the sterndrives do leak and blow gear sets it is typically the upper gear set that goes out. To re-seal and re-gear your sterndrive would run approximately 1500 to 1800 total in parts and labor.
3. One guy says he will pull the engine and check it for about $500, but it's basically like giving him a blank check for what he may have to do. He quoted me around $29,000 for a whole repower, transom etc.
Okay this "guy" you definitely should not be using. There is no need to pull the engine to check it, the engine can be checked while still mounted in in the boat. Pulling the sterndrive is only 20 minutes worth of work. For any competent mechanic he should be able to tell you what is wrong in under 1 hours labor time. Assuming the boat is either on a trailer or already hauled out and up on blocks, and assuming the battery is good and ready to go, it's not much time at all to isolate the engine from the sterndrive and figure out which one is locked up and which one is not. Obviously it could go either way, but odds are it's the sterndrive that is the issue and not the engine.