Going from a 2-handed backhand to a 1-hander, you not only have to change your grip, you have to change your swing path, and your point of contact.
When you hit a 2-hander, the point of contact is even with the front of your body. But for a 1-hander, it's out in front of your body. Basically about a shoulder's width.
And likewise, you have to adjust your swing. The backswing should not go back beyond your rear shoulder. Compare it with your forehand swing. Notice how, on the forehand, you take the racquet back about 1 shoulder's width behind your racquet shoulder. The swing reaches the bottom of the arc directly below your racquet shoulder, and then you come up into the point of contact at the front of your body, one shoulder's width in front of your racquet shoulder.
You do the same type of thing on the backhand, but with one major difference: the racquet shoulder is now the front shoulder. You still take the racquet back one shoulder's width behind your racquet shoulder. But now, instead of being behind your body, it's even with the rear of your body (back shoulder). The swing arc bottoms out directly below your racquet shoulder, which is even with the front of your body, then comes up into the point of contact, in front of your body.
Now, to get the grip. You know how you "shake hands" with your racquet handle for the forehand grip? You can simulate the forehand without a racquet by using an open hand, with your palm for the racquet face, and as you swing through, the palm should be vertical as it strikes an imaginary ball. Then, if you grip the racquet handle with the same hand position, you can keep your racquet head vertical when you make contact with the ball.
On the backhand, you want your palm against the top bevel (those flat edges) on the racquet handle. With an empty hand, make a fist. See how the biggest sections of your fingers (the ones that connect to the hand) form a nice big, almost flat hitting surface. Now, take your backhand stance, with your body turned sideways. Imagine a ball is coming toward you. You want to punch the ball with that flat "face" part of your fist. Take your backswing, down through the arc, and back up until you can hit the "ball" with that "face" of your fist. Your hand should be in front of your body, and the "face" of your fist should be straight up and down, and "pointing" at your intended target. Keep it there. Now, with your other hand, bring the racquet to your fist. Make sure the face of the racquet is pointing straight up and down. Without moving your fist, open it slightly and bring the racquet straight up until the handle is against the palm of the opened fist. Your palm should be on the top bevel. Now close your fist to grip the handle, and you should see the face of your racquet, and the "face" of the gripping hand, are parallel, both straight up and down and pointing at your target. As long as you keep that in mind, you can line up your shots. If the face of your fist is pointing toward the target, the racquet will point that way. (Provided you use the proper grip.)
One last detail: the follow-through. Make sure you hit through
the ball. Do not pll your arm and racquet across your body. Keep the swing going out and up (for topspin. out for slice. it will have a downward trajectory when you make contact, you really don't need to add more.) When you practice your swing at first, exaggerate your follow-through. It will be easier to rein it in later, than to try to bring it out. Besides, in matchplay, the natural tendency is to not follow through as completely as you do in practice.
This would be so much easier to show than describe.