Not a lawyer...Not familiar with Tenn., but here's my $.02
What does your lease say? Then that is what you are bound to because that is what you contracted for. You signed it, you should have read carefully.
I'll guess the thirty days is for eviction proceedings or lease breaking and things like that. But if you agreed by signing the lease that you would be employed or out in seven days, then you should be employed or out in seven days. You'd probably get the thirty days if you stayed after the seven and your employer had to begin eviction proceedings. That's not pretty.
The lease would also be able to answer your question about the partial rent responsibility. In my state, if someone will be moving in right away, it's illegal for the landlord to collect double rent, removing party A from rent responsibility as B is actually living there. Check with your state.
Bottom line: you need to check in your state. Try the state bar association or local landlord/tenant organization. The associations can usually answer most questions for free.
Best of luck!!