A baby can develop, with time, the ability to "empty" a breast in 7-10 minutes. But it does take time for that to happen. During the first few months, it isn't uncommon for a baby to nurse for 20-30 even 45 minutes. I don't pull my babies off, I let them decide when they have had enough and they break the latch on their own. As far as the fuller breast, either baby needs to spend a little more time nursing to drain it, or you just happen to have a great milk machine going ; ) Being only able to nurse on the one side, it is tough for my breast to really get overly engorgered, but occasionally it happens, like when they have hit the end of a growth spurt and sleep a lot. I just let me body adjust on its own.
Some have used cool or warm compresses to help soothe the pain. Perhaps that could work for you.
You don't actually want to pump because breastmilk works by supply/demand. If you pump, you will continue to produce that much. I hope I'm explaining this good enough. The baby nurses strongest when applied to the first breast. I agree with the other comment 7-10 minutes. Start out on the engorged side for a few days, then alternate the starting breast or else your other breast will have a low demand and "slow-down" production. How old is your baby???Less than a week or so??? If you cannot remember which side that you started out on, place a safety pin on that bra strap, then move it after nursing. Best of luck to you. It really is the very best for baby!
I had similar problems with Sam. I now usually only feed from one breast at each feeding (unless he is really hungry or hitting a growth spurt), it was the only was to stop my body producing too much milk and to get Sam to start putting weight on, by switching breast he was missing out on the hind milk and not gaining at first. That said I did express quite a bit in the first couple of months. I would feed from one side and express some from the other side and freeze it. 1, because I wanted to get a bit of a freezer stash established and 2, I think I was scared that my milk would dry up as it did with my first if I didn't keep taking what was being produced. I now see that with my first it was a completely different situation and that is not going to happen now. I rarely pump these days, usually only if I know I am going to be away from baby for a few hours and he may need a bottle whilst I am away. Somedays he does sleep through a feed and I may express a little bit to relieve engorgment and I just add this to my feezer stash which I plan to use in his rice cereal and for thinning out veggie purees and things when he starts solids and also in sippy cups when I want to wean.
Cabbage leaves are good for engorgment, wear a well fitting bra too also hand expressing just a tiny little bit to relieve some discomfort can help too.
Echoing what Andi said, Sam took over 45 minutes to feed when he was tiny, now it's about 15 - 20 minutes, I almost gave up because it took so long at first but I am glad I didn't.
Best wishes with you breastfeeding.
Too much milk is a really nice problem to have. Even if it is just foremilk, I would definitely pump it and save it. If you let baby suck for just a few moments on the leaking breast and encourage him to drink fully from the other, he should get plenty of hind milk. And the less you have the fuller breast be pulled on, the less milk it will tend to produce. That said, if it were me, I'd take all the milk I could get and freeze it. Pump for engorgement (now that my boy is sleeping through the night I have to pump in the mornings or I am walking around like Elsie the Cow) and use the milk either during the day or freeze it.