I moved to North Seattle last April, and was unhappily shocked at the 100% increase in my rent. I've since learned that affordable apartments really do exist, they're just thinner on the ground due to the rise of the mega sized apartment complexes.
I humbly suggest using Craigslist exclusively while rental hunting. There are true deals to be found in every neighbourhood, you just have to search harder.
Also, be aware that the main highway in Bellvue is partially tolled. On the positive side, the metro system is awesome, particularly in Seattle proper. You could consider living further out from Bellevue, as long as there's a direct, no-trasfer route between you and your kids.
Good luck!