When a player shoots a free throw he/she puts backspin on the ball. That spin is in line with the basket, that means the ball isn't ever pulled off line. The "pull the disc to the right phenomena" is actually common, you see it on the part of Pro players in video. The reason it occurs is obvious, if you lift the disc without spinning it, it flies like a wounded duck. To get it to fly right, you have to put spin on it. Spin is to the right so, viola, the disc wants to pull right.
I suffer greatly from this problem. In fact, I have a tournament reputation in my age bracket. I drive about 950, and putt about 750, so I'm an 850 rated player.
Here's my suggestion.
Lift putt, That wrist flick to put spin on the putt, its job isn't just to put spin on the putt, it's to lift the putt into the basket. If the focus in on the spin lifting the disc, instead of just putting spin on the disc, then the force going into the disc is up, not to the right.
Spin putt, this one is harder. The concept is the same, the spin isn't to put spin on the putter, it's to put it into the basket. However, since you're already on line, it's harder to mentally justify that the spin is to lift the disc into the basket. The solution lies below, but you have to use the same techniques for either style.
To get this to work you have to slow down. Go watch Paul McBeth putt. few do it like him. Most bounce the disc up and down to set a rhythm and get the angles right. He doesn't. He looks at the basket, sets his distance and stroke then very slowly lowers the disc to the bottom of his stroke. He pauses there and lifts cleanly and spins the disc into the chains. All players are slower than you think. Most of them have a momentary pause at the bottom of their stroke, Paul's is pronounced. The truth is that you need zero momentum from that down stroke. The lift and distance comes from the flick/spin on the putt, not from the stroke per say.
1) His stroke is slow and measured, never hurried. You don't need a ton of lift and arm speed inside twenty feet. Twenty to thirty you don't need much either. You just aim higher. That is, the point you're trying to drop the disc onto is at the top of the basket, not the middle.
2) Never rush any part of your putt. That's what goes wrong for me. Under pressure, I didn't want too much time to think so I'd hurry. Don't do that. It's an easy thing, lift the disc into the basket.
3) There's a tendency to lift the disc too high at release. To get the disc into the basket on a high release the spin has to push the disc out, not up. On 15 feet and in, my release point is below the basket. The spin has to lift the disc or it won't get in. From 15 to 25 it's about midway, from 25 to 35 it's to the top of the basket (or a little lower). When I'm outside 35 I use a different stroke, one that I've never had problems with. It is meant to put drive on the putter to get it there, and it is more in the vein of a layup. I figured out that if I aimed at the basket, sometimes it fell in.
Practice like you putt. You've read this, I now cause I have, over and over. Put pressure on your putt, and follow the same rules. I don't try and make it autonomous. I try and learn this so that if something goes wrong, I know the parts of the putt so well that I can look my putt and say, that went wrong.
