Switch to an old t-shirt. Paper towels are very soft when wet.
Also try lacquer thinner. Use a wooden tounge depressor to lightly shave/scrape the majority of the varnish off after its been softened by the thinner. If youre hitting wood on wood you've gone too deep, leave a thin layer of varnish. Then remove the existing layer of varnish with lacquer thinner on an old white t-shirt to get the rest. This may take some elbow grease. Use the softest bristled toothbrush you can find to get the stuff out of the dings and waffenamts. I then usually wait for the stock to completely dry a few days (including the remants of the varnish in the wood pores) then the remnants of the varnish will have "cured" hard again. I then use strips of duct tape to "bikini wax" the crud out of the pores. A final quick wipe with acetone and youre ready for stain, oil, etc. This way definitely takes time, but the results are always good.
You could use a chemical stripper. They are quick, however they damage the wood, raise fibers, etc and IMO look terrible. Stay away from steel wool or anything abrasive.