I'm not sure I understand what your issue is. Are you saying the cheapo Walmart amplifier is over-drawing power and overheating? That would indicate an internal fault, or the subwoofer is demanding too much from it, or is wired incorrectly.
Or, are you saying the issue you are having is the lights are dimming on bass hits?
1st, we need to know what the exact issue is that you are having. 2nd, if you are wanting to replace components, then we need to know what you're willing to spend, and what the RMS power rating of the current amp and subwoofer are. Is the amp a monoblock or a 2-channel? How are the channels wired? Is the subwoofer SVC (single voice coil) or DVC (dual voice coil)? Even better if you have the model numbers, and if you know what gauge power wire is going to the amp and what gauge speaker wire is going to the subwoofer. You stated the amplifier is 1000w. Is that peak, or RMS power? Peak power (what is usually advertised) is totally meaningless. RMS is the important figure. My total system RMS power is 1000w, but the peak figures would suggest it's over 2000w, which has no meaning.
You asked what a good sub and amp are... that's hard to say without knowing what your end goal is and what your budget is. I would love to get a Focal subwoofer and amplifier, but I don't have the $3000 to drop on that type of equipment. Infinity Reference is a good line products, and JL and Rockford Fosgate tend be good mid-range lines as well. My current system is a mixture of Kenwood eXcelon (radio, cabin speaker amp), AudioControl (DSP), Infinity Kappa (subwoofers and component speakers), and Infinity Reference (subwoofer 2-channel amp, bridged).
A single 12" Rockford Fosgate subwoofer should be providing plenty of bass, provided its RMS wattage is matched to the RMS of the amplifier, and the amplifier is the correct type with the correct wiring (monoblock is normally preferred, I use a 2-channel amp that is bridged with my two 10" DVC subwoofers setup for a 2-ohm load).
If your issue is dimming lights, have you done the "big 3" charging system wiring upgrade? Google it, there are many related articles.