You might be crowding the ball too much to get a decent swing up through the ball. Usually if we're too close, we can only pull the racquet across the ball and that produces the sidespin you're talking about above.
The one-hander allows for a pretty big swing radius as long as the ball isn't too close to the player. Because of this large radius, I feel as though I can easily generate more power and/or spin with my one-hander, but the catch for me is that I can't be especially rushed to effectively produce the full stroke.
One cue that might help you get better spacing for your stroke is to look at the ball like the face of a clock. If you're a righty, think about extending around the outside of the ball and instead of hitting it in the center (from your point of view), try to make contact at 9 o'clock. If you're swinging lefty, go for 3 o'clock.
Also check your feet. If your stance is either neutral or even mildly open as you swing, that can be making it harder to get the extension through contact I'm getting at above. If my stance isn't closed for my one-hander, I'm much more prone to pull across the ball... and get that sidespin. Closed stance, get onto that front foot before you swing, and don't forget a full comfortable follow through.
It should feel as though you're initiating your backhand swing slightly earlier than with your forehand, since the contact point is ideally a little further out in front of you. Why? Your racquet arm is attached to your forward shoulder for this backhand compared with your forehand (or two-handed backhand) where your racquet arm and shoulder start from behind you.