03 Sunfire electrical problems?

Just bought this car. It will start just fine
sometimes, other times it won't even crank, or
barely tries to crank and doesn't turn over, or i will
try to crank it and it clicks like the battery is dead
but the battery and alternator are fine. Sometimes
when I try to crank it, it does nothing at all. When
driving, gages flicker and wind all the way back.
Seems to happen more when the headlights are
on. Headlights will lose strength then come back
on full strength while driving. No codes, no check
engine light. Around 40 mph when trying to switch
gears it feels like the trans gets stuck but other
times it's just fine, but it also feels like maybe the
fuel pump loses a bit of power and I have to press
the gas to get it going again. I replaced the
alternator and the battery and that didn't fix
anything. Sometimes the anti-theft and check
gauges light will come on together after the
gauges mess up and wind back, and if those two
aren't on, then the low coolant light comes on. The
battery terminals are clean. Any ideas on what this
could be? It has 116k miles. I wouldn't imagine the
pcm is messed up. If it won't crank, I hook it up to a
battery charger and it will slowly try to crank and
after a few tries it will start. Once it's been running
for a while and I park it, I'll turn it off and when I try
to start it it'll either crank right up or do nothing at
all when I turn the key. I've only had it a month and
it's already become a headache. Thanks for your
time