I agree with wpoll and bcredliner that the ignition switch needs to be investigated.
All sorts of weird electrical problems can occur with a failing switch. Check out
this YouTube video for an example of an amazing panoply of issues resulting from a bad ignition switch.
The above shows that replacement of the switch is pretty easy.
Another video link is here.
And if you don't want to spend the money on the switch (or if you want to do an "autopsy" on the old switch to look for failure points), be sure to
check out this video on refurbishing the old switch.
And if it turns out it is the ignition switch, (with apologies to upallnight) you owe wpoll a pint!
Good luck.