Strange behavior from 99 GT

This has happened several times, with increased frequency lately. This behavior is preceeded by Service Engine light by a couple of days.
You get in the car, you turn the key, and either all gauges peg, or nothing at all, either way, the engine never turns and car does not start. Battery is good. You disconnect the + terminal, you jiggle the relays, you reconnect the terminal, and on your first or second repetition of the above, the car starts. The Service Engine light clears. And you have worry free driving for the next several weeks. Than the cycle repeats.
Any ideas? EGR Valve, Cam Sensor, sticky relay, security systems getting activated?
Thanks in advance.

Thanks for the suggestions

The pegging happens at times, other times, you turn the key and nothing. It is definitely electrical but don't think it has anything to do with the battery as it has been checked and I have cleaned the terminals.

As far as the Service Engine code, I have had a couple, with one stating catalytic converter under threshold and the other I can't remember but the AutoZone person checking it recommended SeaFoam which I added to gas. The code has not reappeared. The engine runs fine, once it starts that is.

So, to summarize, the problem is that at times when you try starting the car and turn the key, either nothing happens, as if there is no battery attached, or the gauges peg and the ignition makes a single click sound, but engine never turns. Removing and reattaching battery terminals have resolved the issue a couple of times, yet other times I have had to open up the relays and clean them, and that seemed to have worked. So not exactly sure what actually resolves it.

Any help with narrowing it down to what part of the electrical system could be causing it would be greatly appreciated.