Re: Sucker for the Overhand Right
There is really no reason to get hit with right hands because you can see them coming for a mile...unless, of course, you are square to your opponent. Then you are considerably closer and thus easier to hit with any straight punch. I think that that may be the case, as you mentioned that you were getting hit with a lot of jabs, too. (Catch 'em in your right glove, it is easy to do. And, you can pivot off the catch to cut distance and jab back, or parry it to create an opening to get close and work inside.)
I don't think that keeping your hands up is the answer; you mentioned that even when you blocked his right hands, they were damaging. In fact, keeping your hands up may well be part of the problem, as it is easier to avoid something when you can see it coming. Get yourself more sideways because that will make you less of a target for right hands and jabs, you'll be able to see them coming for a longer distance, and you can hide behind your shoulder.
You said you often knew he was going to throw the right hand but still got hit with it, so I guess you stood there and waited for it, gloves up? Get under it and hook his body. Slip outside it and counter with your own right hand. Slide to your right, away from his, and beat him to the punch with your right hand.