Does running a dehumidifier with an AC unit save money?
A dehumidifier is about the same as a window AC unit with both parts in the house, for most purposes. They draw just as much power as an A/C unit of similar size (or more if they happen to be old and inefficient.)

Given that the required energy labeling on A/C units tends to promote a degree of of paying attention to that, and it's much less clear on dehumidifiers, I suspect that in many cases the A/C of similar size will actually use less power - and it doesn't make your house hot.

Sure, comparing a room dehumidifier to a whole house A/C, the whole house uses more energy - but it also removes more water from the air, and cools the house. It would only be ANY savings if the whole house was LESS efficient (power in to water out) than the dehumidifier. If you are running the A/C as well, you are also ADDING the heat from the dehumidifier to the A/C load.

In short, run the A/C, don't bother with a dehumidifier unless you have a job that a dehumidifier (alone) is suited for; Which it sounds like you probably don't.

If you want to get "something like that" without the downsides, when your water heater wears out, consider a heat pump water heater - free hot water with a little A/C, or free A/C with your hot water. Still rather expensive, but sometimes with large rebates available too.