I went on this one Beatles website, and found Ringo's and 2 Paul addresses posted there. I asked the site owner if they were accurate, and they said that they got a reply and an autograph from Ringo! They said that if you use decorative stationary and politely ask for an autograph, he will send you a picture of himself signed! So, as far as I know, if you use this address, which has been mentioned before on here, he MAY reply:
1541 Ocean Ave #200
Santa Monica CA 90401
I'm not so sure about Paul, that site owner said they wrote to him using one of these, but didn't get a reply. Well, if you wanna give it a try, here's Paul's:
7 Cavendish St.
St John's Wood
London W1 England
These aren't guranteed, but it's all I've got!