Did you check the battery cables? Sounds like a issue with poor grounding. The point were the negative battery cable is connected to the body of the car may be rusted or corroding and causing a poor ground.

 

Check your ground wires from your block to your chassis and your grounds from firewall to block etc. More than likely (as already stated) its a ground problem, however it is not limited to the battery terminal. The battery ground is only one wire from battery to the fender well. If the ground from the block, chassis and firewall got disconnected somewhere, the car wont transfer common ground to the rest of the car. Im not sure where the harness may be grounded for the dash equipment, but if its grounded to the fire wall and the wire that grounds the firewall doesnt have good contact, these symptoms can happen.

I just fixed my buds truck who had the same problem. He had air bags and when he locked up, it stretched the ground wire from the block to the chassis. His gauges and such were going crazy. Found that they got their ground from the same transfer wire that was going from the block to the chassis.

 

You can bump the thread all you want, this is going to require leg work on your part, sorry. Very hard to diagnose exactly which cable in the car is having the problem when you are talking on a forum over the Internet. It's a cable issue, battery or voltage regulator. Since you have replaced the battery and the alternator, it's a ground issue (probably). Get a multi-meter and start testing voltages and resistance on ground cables or take it to a shop.