Just sharing my thoughts....
I agree with 88_keys on not to practice separate hands too much; it'll make it even harder to combine since u r so used to play them separately. I also think it takes u longer to learn each of the separate hands until fluent, than if u just learn them combined.
Why not try practicing one hand a little bit (not too much, just to get familiar) and then start on the other hand but playing them together. Eg. practice RH for a little while, then start to practice LH, but play it together with the RH. Bcoz u r quite familiar with the RH, it's like just "adding some harmony or melody" to the RH.
It can also work the other way round, I think (i.e. LH first then RH-LH combined). You can divide a piece into several parts, and practice as above for each part.
It also helps to listen a lot, from time to time, to the recording of the piece u r learning, to get familiar with the sound of the LH combined with RH.
Hope it helps. That's how I learn and I think I pick up new pieces quite OK...

Happy playing! (I certainly am!

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