Acid shouldn't damage the plastic drain but it's hell on copper pipe.

I second (or third) the suggestion to clean out the drain by probing and
flushing with a plunger. I have long hair and have to clean out the sink
drains from time to time. You can just use a thin bladed knife to poke
around, or a bent coat hanger to get right down into the trap. Then fill
the sink half full of water, pull the plug and quickly put the plunger
over the drain. I get the best results pulling up on the plunger to suck
stuff out of the drain instead of pushing to try and force it through the
pipe into the sewer. Pick up the sludge that gets sucked back up inte the
sink and toss it in the garbage.

I've replaced a soldered solid brass P-trap with plastic and compression
fittings. It's straight forward and easy to do. All you keed is a hacksaw.
However take a look at the pipe too, the sludge built up in thin chromed
copper pipe promotes corrosion and those piples can become thin and weak.
I've had pinholes in those pipes. This house was built in 1964. I too have
a double vanity in the upstairs bath. I replaced both sinks and one of the
P-traps on those last spring. The faucettes were soldered to the water
pipes. They were removed by melting teh solder with a torch from teh
hardware store, and installing new shutoff taps with compression fittings.
No need to solder oor glue connections these days. All the fauwcettes in
the house now have P-traps and shutoff valves installed with compression
fittings.

Drains are funny. I used to rent out the house next door to a tennant.
The upstairs toilette was slow draining. I tried probing and
plunging but found nothing. Eventually had to remove the unit and found
nothing but a small metal button like a man's cuff link or something.
Frustrated, I reinstalled the unit. It worked fine after that.

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