Do you have a ceiling fan in the bedroom? We have an elderly central air system and keep the temperature at 78, and with the ceiling fan going at night, it is *plenty* cool for sleeping.
While the kids are probably fine with sleeping in your room for now, that's not a long term, big picture solution.
How long is your cooling season? We have maybe 9 weeks in the summer that we run the a/c, so while it does shoot up our electric usage, over the average of the year it's not too bad.
For me, if the central a/c died now, and I didn't have the cash to replace it, I'd buy a couple of window units. My house isn't that big, and we don't need to cool it for a long stretch. The central a/c is nice, but it's definitely a big investment. (I plan to replace it next summer, along with the furnace.)