Thanks for the tip about dragging your feet. I have always felt rushed into signing the lease (afraid I would lose the apartment if I didn't hurry) especially when moving from out of town. I didn't realize the application fee actually held the apartment.
What do you think is a better price estimate? Do you think it is realistic to lower our housing costs below $1235/mo or should we just try and break even?
I know for our area (near Clay and BW8), we looked and single family homes were >$1300 for the most part. And those were not in particularly great areas. I didn't look too much into townhomes, there may be some deals in that market if you just want to not have upstairs neighbors.
I'd say start researching whenever you want about how much places 1-2 months out from today's date are charging. That gives you an idea of the range you should be shooting for, and will let you know what a good deal looks like. Narrow it down to 5 or so places you like, and email/call them to ask how much notice they require, that lets you know when to really start looking. Some places have really updated online stuff, some don't. The updated ones usually list individual apartments prices and availability. The non-updated list a range for a particular floor plan, or say "starting at $XXXX." Try to find a time when you'll be at max availability for your top picks, within a week after move out notice would be required, and go and try to visit them all in one day. Many places give incentives for applying within 24 hours of touring, so try to take advantage of it. If you like one, fill out the application and pay the fee.
Once they send you the lease, wait until they follow up. Wait a few days and respond saying you don't check that email very often or something and will try to take a look at it this week. Ask a few questions if you want later that week, stuff like fire alarms and sprinklers, year of construction, etc is usually required for renter's insurance, so those are good candidates. Stall until 1-2 weeks before move in, then sign the lease and get utilities and insurance ready to roll.
edit: Just remembered you weren't really looking for apartments. House rentals are probably harder to come by, so I'd snap up one if I found one I liked at a decent price. The above is a good way to go for apartments if you decide to go that route though.