If I sign an exclusive contract with a library, once that contract ends, am I allowed to take those tracks and submit them elsewhere? I would hope so but am really not sure.
Yes, once the contract is over, you can remove your songs.
Be sure to always read the duratiion of your agreement. Many Exclusive libraries these days want to keep your track in perpetuity (Forever). There are some good Exclusives that have 5 and 3 year clauses.