Well yes, she will need to have a lot of the milk removed somehow.
It is impossible to say from here and now how much milk she is going to give.
Little bitty calves generally prefer only 1 or 2 quarters to nurse from.
That leaves the other quarters to swell and possibly become infected.
At the very least the unmilked quarters will dry off, though not immediately.
It is a delicate balance to keep a nurse cows udder in good condition.
Too many calves and they just suck the weight off the cow and the lowest calf may not get enough milk.
Too few demands on the milk supply can end you with an uneven udder and mastitis.
Also, calves will avoid the sick quarters if at all possible.
So you will be playing a lot by ear here, especially since you are afraid to get under her yourself and look.
Perhaps you could work on that at least some.
I find it difficult to believe the calf isn't nursing her (and stealing colostrum from the unborn one).
When you say her udder filled up, I bet that is why. Just a guess.
It is really much better to be SURE the calf isn't nursing her. No good will come from that.