I changed status from F-1 status to permanent resident end of June. I've been with the same employer from the beginning of the year but I might change soon. Should I receive two W-2s(one for the months under F-1 and 1 for the months under permanent resident) from my current employer and then another W-2 from my new employer or nothing if I become a contract worker.
Should I use 1040EZ-NR with the F-1 W-2 and use turbotax/H&R block for the remaining months?

If you are married, you can and probably should file a joint return.

If you are single, you will need to file a dual-status return, which is NOT a return for amateurs (and NOT for a fair number of tax professionals, for that matter). Get professional help from a tax pro who has successfully filed dual-status returns in the past.