Drano will not hurt the plumbing, but might not be good for an RV septic dump station--I think it is a caustic soda material with a high pH.
Consider using one of the barbed hair retrievers to run thru the traps to see if a hair plug is present--very cheap item and easy to use. I would also pull the traps and manually clean them if the barb gadget doesn't work.
I think you will find that you have hair built up, along with soap residue and using the barb works well for us.
2016 Tiffin 40 QBH
2015 38RSSA, traded
2005 TK3 #1869, 10 yrs of memories,
2017 F450 KR--one more Ford is it
2009 F450 4x4-died; 2010 F450-retired