What is the actual bore and groove diameter?
I am surprised that .010" is too thick.
Dark stained green or purple heart was also used. Ramin would also stain well.
Rosewood as well, but unlikely stain well due to the oil content.
Osage makes a WONDERFUL rod.
Can you measure the rate of twist? (patch on jag) .620 is between 19 and 20 bore.
Might have to use a card or felt barrier wad under the patched slug if you can't even use a .0210" patch

Ballistic Products sells a dry 20 bore X 1/4" felt wad that measures about .626" in diameter. It's difficult as it compresses very easily.
Soaked in BP bullet lube like SPG compressed & hardened might help. Lyman BP Gold as well as well as 60% BW:40%
Vaseline is also a good Black Powder bullet lube.