OK, sounds like engorgement.
Just before giving birth all mammals begin to produce "first milk" it is called colostrum. The baby needs this, after a day or so the actual milk begins to come in. Sometimes they produce too much. This causes engorgement.
As soon as she kids try to milk her just enough to be sure her teats aren't plugged. Once you know they aren't, get babies nursing. She will still be very full, if she's so full they can't nurse, milk out enough to lightly soften her teats. Don't milk her out all the way. Maybe a couple of cups worth.
Once the baby is nursing well it will regulate the milk supply. But if you see she's staying regularly full, you can continue to milk after baby nurses and increase the amount you take a little each time.
It is supply and demand at that point. What gets taken out by you or baby gets replaced by the goats body.
Also, is this a dairy goat or a meat breed?