Okay so the 13V was probably more like 6.
It's the type meter that half-wave rectifies AC, then reports average multiplied by a scale factor. That's okay so long as we are aware of it.
DC multipled by that scale factor is more than the DC was. If you reverse the leads it'll probably read zero not peg downscale.
One must know his test equipment - just like people what it says isn't always what it meant..
That sounds like either an open spot in potentiometer near the end, or there's a small signal making us charge the capacitor.
Check pot with ohm-meter over full range of adjustment.
You are getting close i believe.
What are values of C1 and C2 now?
I may have not made this clear - opamps generally dislike large capacitive load because they can't deilver enough current to change the voltage across capacitor quickly (i = dv/dt)
so they can't make their feedback and burst into oscillation.
So we picked a speaker driver opamp with fairly hgh current drive.
That's why i am nudging for a smaller C1 & C2, with maybe a very few ohms in series with amp outout pin. 1uf C2, 1 meg pot, and 0.1uf C1 is what my intiution says would keep LM386happy in audio range. Give pin 5 wire a few turns around a ferrite donut (from junk PC power supply) perhaps.
old jim