If I understood you correctly they are hosted on the same server, so there's no easy DNS change that can be made.
Depending on how you have your domains setup (is one a main domain and the other an Addon) you could ask your host to change the document root for the main domain, but you would still need to change the site URL for WordPress since you are changing it from .net to .com. You would also then have to change the folder the other domain points to as well. This isn't normally a recommended practice since it involves changing the normal way the server handles files.
Moving files isn't as hard as you might think and if you back everything up first you'll be fine. Yes, you should be able to leave the database alone when moving the site files. I recommend backing up everything when making major changes like this because in the event something goes wrong, you want a backup of everything, just in case.
Hope this clarifies things a bit.