Yeah.. Do what the others are suggesting first, but then I'd check the wiring from remote to engine.. Using multi-meter to check continuity.. I replaced my ignition switch when I found that the resistance in the switch when I turned it to start would sometimes be good, but then sometimes would jump to 250+ ohms resistance... Guessing the insides if my switch was corroded somewhat, and I too was having an intermittent issue where it wouldn't even try to start. Sometimes... Not as drastic as yours, but turning the key sometimes worked and sometimes not.
No issue since replacing it...
Now, I'm not saying that's what your issue IS, just that I had that issue, and it might be worth checking on yours... Still learning myself, so for whatever that's worth... LOL!!