Sadly, the fault very likely does fall on the shouders of your "cheap wireless mic setup" my friend. When I started out in commercial and public access video production some 20 years ago I did the same thing - picked up a multiple wireless mic system. Some RadioShack-like thing, hoping it would get me by. From day one I had many of the problems associated with inferior and/or cheaply-made electronic gear.
When you under invest in such equipment, while it is possible to occasionally find a gem buried in all that coal, it is usually so full of inferior circuitry, components and elements as to make the actual investment a total waste of money - a black hole for your hard-fought dollars.
So, yeah, you wanted to hear answers "besides..." but the only way you are going to be able to combat the huge number of possible reasons technically or signal-wise why your system is experiencing this problem, the major probable reason is the LACK of quality of your "cheap wireless mic setup."
Even with some systems that cost you MORE you'll have signal issues, and occasionally those systems in the $1K or higher range will experience a few themselves. There IS NO PERFECT wireless system, though some of the really expensive (Letrasonics (SP?), for one) might come close.
Sorry, I have no specific answer or solution for your existing system, but a few of the answers you don't want to hear, in addition to higher-priced and better quality wireless include hardwired mics to a mixer, a hardwired mic pointed at the PA speaker system "sweet spot" and connected directly to your camera, or placement of several self-contained, free-standing digital recorders similar to the Zoom H2.
