2-3 night backpacking trip in southeast Utah

I’m flying to SLC with a friend on March 12 and spending 6 nights in Utah.  We’ll rent an SUV and likely head toward Moab.  We’ll likely sleep in the SUV or car camp when we’re not backpacking.   I’m looking for advice on premier 2-3 night backpacking trips in that area (Canyonlands, Dark Canyon, Grand Gulch) or anywhere else in that region. As well as spots to car camp between trips.  Preference to places where permits are easy to obtain or aren’t needed at all (not sure how realistic that is).  However, willing to do walk up permits or take any means necessary to obtain permits for primo backpacking trips.  I had looked at permits for needles district last week and it looked like nothing was available for the time we’ll be there, which surprised me given the time of year.  But ya live and ya learn I guess.

Any advice at all would be helpful.

Thank you!