OP: may we have exact information on how many kids you have? Also, need clarification if your flights are mainline (either Airbus or Boeing) or if these are United Express flights (on commuter aircraft). The strategy might be different given the different aircraft seat maps.

Other thoughts - not sure if it's too late for you to do these things.

1) What sorts of seating arrangements are you willing to accept?
If you're sitting in a 737/A319-320, economy seats are arranged 3 by 3; if in UX aircraft it's either 1 by 2 or 2 by 2. So how will you group yourselves with your kids? Note too that some flexibility (ie you and your wife sitting far apart) at least officially at boarding is better than nothing. (See point 2 next).

2) Be prepared to negotiate seat swaps with your neighbors once boarded.
You and your wife need to be booked in aisle seats (suggested by another poster), also as far forward as you can within E-. Doesn't matter if you're in different rows, many fliers find aisle seats preferable to window. If you've got an aisle but can swap with someone in a middle seat so you can sit with your kids, someone will always be willing to swap with you, but if you're in a window/middle seat, good luck swapping with someone in an aisle seat. If you're separated, but your family has got all three seats across, you can probably get a row of three seated closer to your spouse to swap your three seats with theirs.

3) Further point 2, if you are indeed stuck in a window or middle seat (so lousy bargaining position), willingness to offer drink chits or purchase a BOB meal might change said person's attitude.

Finally, hope for the chance that there are at least a few elites on board who through CPU may get moved into the E+ seats at the gate.