I find myself in the same situation as the OP describes all the time because I fly the D11 a lot. If allowed, I will try to gain separation and keep my speed up, all the while try and chip away at my opponents energy advantage when I can. Speed is your friend in this situation. Sacrifice a little speed for altitude when you get an opportunity, ie, the opponent sacrifices some of his speed to gain a hard shot or an HO. Don't give it all up - you need it to keep your AC responsive. I find the majority of AH pilots will get impatient, do exactly this and that is when I try to gain an angle and altitude. Once I roughly have equalized energy states, I get really aggressive and start using flaps, trim, etc.
Its worth stating in this discussion, at least for the P-47, when you need to reverse quickly after your zoom, use gravity to help get your nose around. Chop your throttle a bit to limit engine torque, initiate your turn and firewall it again. This will let the torque and gravity work together to get your nose around. This works really well with the P-47.