We tend to use vrbo and using the map click on Florida, then Central Orlando, then Kissimmee in the list that comes up then back to the map for west or central west. Far West can start bringin up some of the longer drives again.
My list would be
Formosa Garden & Windsor Hills (while top of the list, I've never stayed there as always too much for us! They look the nicest though)
Windsor Palms & Indian Creek
Rolling Hills (plus any others around livingston road)
We've stayed in Windsor Palms and it was really nice, gated and manned 24/7 so felt nice and safe and welcoming although does add a little to the journey as it's the furthest of the communities and you have to wait for the gate etc.
Rolling Hills we've stayed at twice, it's the cheapest in that area (you can definitely get lower than 500 per week, we did 350 per week for a 4 bed there) and is a matter of seconds to Formosa and Windsor which are twice plus the price. What you'll get though is an older villa, no gate or security and a mix of residential and rental which does change the feel of it. However, as a cheap alternative to a hotel and so close to Disney it's a good option.
Finally, use your 3 weeks as a bargaining chip, this saves the owner a lot of money in cleaning fees and your small family will also get a discount in a lot of cases. Be up front in asking if there are any discounts for long stays and or small parties. For the owner, this means fewer days empty, fewer cleans and less wear and tear.