I practice heavily using LSAT CR and RC questions. I do strongly recommend LSAT CR, as GMATNinja does, because LSAT CR is much harder than GMAT, so it will prepare you well for the GMAT by making the GMAT questions seem easy.
You are correct that there are some LSAT question types that do not appear on the GMAT, mainly the "similar reasoning" type questions. I personally skip those, since it won't really appear on the GMAT (or at least hasn't appeared yet). I mainly focus on LSAT assumption,strengthen,weaken, and paradox questions since those are the main question types on the GMAT.
It is normal that your LSAT CR accuracy is lower compared to your GMAT CR accuracy, since the questions are denser and trickier. 60-70% is a pretty good accuracy I think for LSAT hard, especially when you consider that they are harder than GMAT questions.
If you are trying to improve RC, I also recommend LSAT RC passages.
GMATNinja made a great post about why you should use LSAT RC and CR:
https://gmatclub.com/forum/experts-topi ... 39365.html For SC, OG is the best, but if you are running out of questions I really recommend Manhattan SC book.