Not emptying the breast

Hi mammas.

My son is almost 11 weeks old. By 2 days ago he was waking up twice per night to eat.

The previous 2 nights he went bed before 9 pm and woke up at 5 am to nurse. So being so long for the first times, my breast were sore, full and hard. He emptied one, and only ate a little from the second and was full, fell asleep and didn’t want to eat more, and my other boob was still a little sore. What to do in this situation? I am afraid if I will pump the rest it will increase my milk supply during night time and I will have to do it all the times. Or that’s not true?

I am exclusively breastfeeding him.

TIA