I just installed a new sink that has basin lower than the old one. The tailpipe from the trap meets the bottom of the sink so closely that I can't fit a new drain in. I'd cut it, but the drain for the dishwasher also meets at the tailpipe area above the trap. I'd like to lower the trap, or rather make a deeper curve to it so it can allow for more room below the sink, but can still go up to meet the drain coming from the wall. I'm afraid gravity will play too much of in issue. Any thoughts?

Traps are designed to work with a standard wall piece and a J bend. If you adapt extra pieces to the J bend it will affect the efficiency of the P trap. I have resorted to installing a S trap in cases like yours but that is a code violation and might haunt you later if you sell. The correct fix is difficult depending on your cabinets and the drain inside the wall. The horizontal drain line needs to be lowered. The only good piece of news is that all the destruction you would be doing is out of sight. Are you on a slab or is there a basement? If you went with a Strap you might need an AAV or a Chicago loop vent to avoid siphoning the water from the trap.

You may have a clean-out below the drain inlet. Inspect the wall inside your cabinet. If positive, use clean-out as drain inlet and original drain inlet as clean-out.

Turn the trap around backwards. You will probably have to use a longer tail pipe from the sink to the trap. In other words, connect the longer part which normally is the inlet, to the trap arm. Connect the shorter part to the sink. The tail pipe will only go inside a short ways but will seal. Simply means you have a deeper trap, and it gets you back up to the stub out from the wall.

Turn the trap around backwards. You will probably have to use a longer tail pipe from the sink to the trap. In other words, connect the longer part which normally is the inlet, to the trap arm. Connect the shorter part to the sink. The tail pipe will only go inside a short ways but will seal. Simply means you have a deeper trap, and it gets you back up to the stub out from the wall.
Is your old trap Plastic or metal? Good thinking Harold,unfortunately it won't work by reversing the "J" bend. The trap tail connects to the high end with a ground joint in PVC and with lip and rubber gasket on a metal "J"bend while they both use a compression fitting to pick up the sink. The two connections aren't compatible. However, your thinking was bang on. You could build a running trap,(see image) out of 90's. Come out of the wall with a 90, drop to the elevation that you wish and build a running trap out of street ells and turn up with a raiser and a compression fitting to pick up the sinks branch tailpiece. Your trap would differ from the image I put up with the 90 coming out of the wall connecting to a drop instead of directly into the return bend. This puts then trap low enough to make your connection back to the sink with a compression fitting. Good luck. Tom