Sword,
It's always good idea to stay with a valid H1 as much as possible.
If your working with EAD and your I-485 is denied, you will
be without status immediately. But if you have a valid H1 that time, you will be automatically reset back to H1 (non-immmigrant) status. So, if you want to stick with current employer, ask them to extend H1 and and mention that you are going to pay for it (explain them that you don't want to take risk).
But if you want to change employer then either transfer
your H1 or use your EAD. It's always safer to transfer H1. But nowadays very few companies are willing to sponsor H1 visa. So if you get a stable and good offer, do not hesitate to use your EAD (that's because I-485 denial is unlikely, if you working).
As your EAD is expiring on June, you should apply for renewal immidiately, as it's taking longer time. You can apply in your own (to save attorney's fee).
You need I-485 receipt (ask from attorney, if you don't have with you), photocopy of current EAD (both sides), current I-94 (both sides), two pictures (immigration type).
You can see INS site and
http://www.immihelp.com/aos/ead.html for more details.