Hi guys, I am having a problem with my amp and music system.

I have a Sony XM-1252GTR 340 watts RMS x 1 amp connected to my Sony XS-L103P5 sub with 330W RMS, this is all in my 96 Accord.

When i first got the amp and sub I used to listen to it alomost full blast for a good hour before my amp started to give any problems, and even cutt off the bass.
The only problem actually was that my lights were dimmin when i drove at night with the music up.
I was recommened to get a capacitor, so i got a 3. farad power accoustic off ebay for about 40 bucks+ 20 shipping.

Point is that after i got all of this installed at the local music shop, the cap seems to have improved my bass responce and it seems to hit harder, but now my amp is heating up and cutting off the bass in about 10-20 mins tops with the music blasted. And also at night my lights still dim. The capacitor lights up and all and has this display that shows the voltage and stuff. Im curious because when the music is blasted and i look at the cap. it shows the volage dropping usually around 10-12, so it that a problem?

I also took my car back to the shop where the putt the cap. and he says i need to get a new battery because after adding the cap. he says its draing too much from my battery and thats why i need a yellow top.

any suggesttions guys? Could it be a bad amp, capacitor or do i need a battery?
Iff i need a battery any suggeestions as to which ones? Im on a budget so lets keep the battery cost under $100

Im a noob, so any help appreciated. Thanks in avance guys.