Converting 24VAC to 5VDC - Need transformer demystification
You can't rely on using resistance measurements to get turn ratios, because they almost always use different gauge wire for the two windings. The primary winding is usually several hundred turns of some fine gauge stuff, while the secondary is a few dozen turns of something heavier, to carry the necessary current.

Probably the easiest solution is to get a 2nd wall-wart that supplies a few hundred mA of regulated 5V, and you're done.

However, if you really want to DIY it, you may be able to find transformers with 24V secondaries that are center-tapped, in which case you may be able to use one leg to the c.t. to get 12V AC for your diode-and-7805 approach.