Don't believe for a minute that UA will ensure that you will not be separated on the flight from your children. I was on a recent flight SFO-SEA and had a boy about 6 years old sitting next to me who didn't speak a word of English. I asked the FA where his parents were and they had no idea! Finally one of them found his dad about 10 rows back. They made no effort to seat them together.

OP, check every day and if any seats come available, grab them. It's likely more seats will open up closer to check in, and if they don't, plead with a gate agent. Personally, I'd just pay for E+ seats.