2002 Dodge neon sxt lights turning off and on, fuel gauge going to E,the binging wont stop

i got in my car yesterday and when i turned it on the lights and radio
starting going in and out, the fuel gauge kept going down to E ( like
it does when you turn the car off) and the normal 2 or 3 bings you
hear when you get in wouldnt stop. I turned the car off and back on
it did it again. the next time it was fine. i took it in to the shop
today and they ran tests on the alternator and it was bad, had that
replaced ($530) went to pick up the car and it would not start (they
left the headlights on and the battery died) so they gave me a new
battery. Drove the car tonight. stopped at the store - when i got
back in it did it again. the next time i started it to. i dont know
what to do, i cant really afford any more work but I know it would be
more costly if I have to get towed. Any ideas? may plugs? wiring?
any help would be greatly appreciated.

thanks