Hi wlooo,
I think from how you explain your shove-it's, your problem is that you pull the sail to the back.
To get into the shove it position, you need to carve upwind and jump with the board, at the same time, you push down with your mast hand and pull up with your sail hand. This gets the sail down.
When you just pull it to the back, the sail will also go down horizontal, but you won't be able to get the wind push against, for it to come back upright,... the mast is gonna point straight downwind...
If you do this mistake strong enough (pulling the sail back), you will probably get wind from below, which will push you over the sail and let you do a Cheese Roll.
Yes, of course you can do a cheese Roll

- why not... you tried, it almost worked... try it several times more and you'll probably do it... exaggerate the movement by looking with your head to the clew or over your back shoulder and once you are around, come under the sail and sheet in to get it back upright.
Now for the shove it - its really like working with a shovel,... the movement is the same - you push the sail down AND FORWARD with the mast hand and get over the sail. The tweak will come automatically once you are more comfortable.. it basically is not a rotation, but rather you going more radical. You push the sail more down and more aggressively forward, at the same time, just push the back foot up and back (extend sail hand and back foot)...
So I would say the difference is the take off - for Cheese Roll, pull the sail back and look back - for Shove it, push the sail down and forward, to come back upright, push out with the sail hand, to tweak it, push the back foot up and back at the same time
Be careful with shove it landings once you can do them... focus on landing on your tail!! You can land them full power planing but the danger is that you come down very flat - i broke a few boards with that already!!
have fun and let me know how your next attempts go!
Rossi