The .015 thick patches shouldn't be doing that with a .490 diameter ball and a 60 grain powder load.
As the others have mentioned, a new barrel will have very sharp rifling grooves. These can cut the patch during loading or when the gun fires.
Another thing that can cut the patch is the sharp edges of the rifling at the muzzle.
Using a small piece of the black "wet/dry" sandpaper and using your thumb to rotate it on the muzzle crown to round off those sharp edges can really help to stop patch cutting while your loading.
You didn't mention what kind of lube you were using or how much you were applying but IMO, whatever your using should thoroughly dampen or cover the cloth patch.
When building a new rifle, I always get a piece of steel wool and wrap it around my cleaning jag.
I then run it up and down the bore about 10 times, remove the steel wool and jag, re-position it on the jag and then repeat the process. It may take up to 50 strokes to do the job.
The steel wool is about the same hardness as the barrel steel so it won't really wear out the bore but it will round off the razor sharp edges of the rifling grooves.
This will not hurt the barrel or its accuracy but it will greatly speed up the "break in" process.
The alternate method is to shoot over 100 patched roundballs thru the bore.
You said you were using 3Fg powder but if it is the Seven7seven powder it has a reputation for burning patches, even in moderate powder loads.