Honestly, boy that is going to be tight.... Keep in mind that May 25th is Memorial Day and the last business day of the month is the 29th. If your loan is approved and you can get a bilateral contract by Friday you may stand a chance but you would need to be going fast. Appraisals are backed up right now with the refi boom so that could take as long as 2 weeks to come back.
I am sure that many people who would like your business will tell you it can get done by the 29th but you have to understand the overall market right now. If I were in your shoes, I would plan to be a homeowner in June, not May.
Renting on the other hand - I don't see why not. Good luck finding your new home.
Bank of America Home Loans