I had pretty much the same issue with my .308 a while back. As you mention, putting the tip off-center does help a bit. Still, I ended up cutting them all the time after almost getting one stuck in the bore.
Came across
these a while back. The theory behind them is that the shape is less prone to bunching up, but also offers more surface area. To me it seemed to make sense so I gave them a shot. In addition to .308 I also bought some .223 of the same brand; as my normal .223 patches were on the opposite end of the spectrum in always feeling too small.
I've only had one cleaning session with them, but they're excellent. Nice snug fit that doesn't bind up or feel like it's about to get stuck. Definitely worked as advertised, and they're probably all I'll use from here on out.
One thing I noticed is that the price right now is about DOUBLE what I paid a few months ago!?!? I know Amazon flexes their prices, but this is a pretty extreme jump. Then again, it's possible that I just got lucky and caught them at an extreme discount. Just to be certain I checked the order history and confirmed that the 200 Pack (.223) and 120 pack (.308) were only $4 each at the time I found them. I do remember at least one was an "add-on" item; which you can only purchase if all the other items in your order come out to more than $25. So, maybe that's why the price was lower. Might be worth putting $25 in your cart and then trying to search for the Boresmith patches. If not, and you can wait a little, maybe just keep an eye on the price to see if it will go back down.
Even at the higher price I would probably buy them. They just fit so well and really seem to clean the bore better than their square and circular brethren.