Ghosts in my plymoth Neon!

Hello. I have a Plymoth Neon 2000 model with 108,000 miles. The problem I am having with my car is that not one single mechanic I have taken my car to can figure out exactly what is wrong. The problems are varied so bare with me. First of all the headlights and interior lights dim and flicker. I notice a lot more problems when the headlights are on but they don't have to be for me to see a nice light show inside my car. By this I mean that all the lights on my console flash (seatbelt, airbag, brake, check engine, ect.), The gas gauge swings all around, speedometer swings, temp gauge drops to cold, the left turn signal is lighted up without being on (if I turn on the right turn signal the car stalls and imediately goes dead)(also the headlights are dim without even being turned on) and while this is happening if I turn the headlights on it will also stall. When trying to test other things on the diagnostic machine I tried to press the accelerator and it couldn't stay steady enough to test anything. When we checked the spark plugs there was oil pooled in 2 of the sockets so we changed out the wires and plugs. It hasn't been a problem since. The alternater tested ok but when we pulled it out it was faulty so it was replaced and within a week it began to go bad again. We changed the computer. We also got a new battery. With all these things being changed without anything being fixed the cost is adding up. I have had the grounding wires for the battery checked and everyone says they are fine. But I was wondering if it is possible for one of the wires to be shorted out or something. Any ideas would be welcome. I am a woman so if some of these explainations are vauge I appologize. Thankyou.