Especially when you are trying to adjust to jet lag, etc. I would pick one or the other, and do it up right. A reasonable hotel that I stayed at was the Hotel Columbus right down from the Vatican. Centrally located, you can walk to alot of things from there, the Forum, the Colosseum, etc.
Here's their website:
http://www.hotelcolumbus.net/index_en.htmYou can rent an apartment too, which gives you more room and probably laundry/cooking facilities, but I can't recommend any specific place in that department.
Go to TripAdvisor, I have been using them for awhile now to find little gem hotels and get advice on restaurants, etc.
May is a great time to go there, not too hot yet, and not too touristy, lots of daylight and sunshine. Have fun! -48