I am not there, I don't know what is wrong with your car. A few minutes with a voltmeter and your keys and I certainly WOULD know, but maybe this will help.
There is a ground strap from the engine to chassis/frame. Mine went open and I had a similar symptom, I did not waste time looking for a damn broken strap - I could measure almost 5 volts between the engine block and battery negative post (I told you that a voltmeter will tell you what you have) - so I KNEW it was open/broken/gone.
My answer -
Go to the store and get one of these:
Now, cut that damn yellow splice off there and crimp on a ring terminal lug of size that fits the battery cable ground bolt just under the fender near the battery. Next, remove the existing battery ground cable and throw it away. Grab your new cable, fasten the clamp to the battery and then secure the ring terminal with the battery ground bolt at the fender that you just removed in order to get the battery ground cable off.
Lastly, run the HEAVY lead to the engine block, find a big-**** bolt and put it under the bolt and tighten. Ideally, you should use the bolt the bad strap is using - but I sure didn't. I removed one of the alternator bolts and passed it thru the lug of this cable and tightened it down. Problem solved.
Were I there, before I did ANYTHING I'd go looking for voltage drops on the positive side when you turn the key or voltage gains on the negative side when you turn the key. Ohm's and Kirchoff's laws will tell you what you have. BUT, given how you descibed what is happening, I'd ALMOST bet the ground from the engine to the chassis/frame is missing/gone/bad/etc.
and my fix above would get you around that for about $15usd and maybe 30-45 minutes of your time. Even if I'm wrong here, would $15 and less than an hour of your time be worth it to KNOW that the grounds from engine to chassis/battery are GOOD? Because doing what I'm telling you to do WILL give you THAT.
If you do what I said and NOTHING changes, well, that means we KNOW it is on the positive side, so then open the little box near hte battery and inspect the tightness of the nuts on the 200amp fuse in there. But I don't see that as the issue, I really dont.