For a badly plugged up tank, try liquid Drano. Our owner's manual says to use it. Needs to be the version that does not attack metal. We had a badly plugged black tank last year and used it and the tank ended up looking like new inside. If you do use it, NEVER dump it into an on-site septic disposal system or you can do serious damage to it.
If your black tank is a problem, you may not be using enough water in the tank. You should put in 4-5 gallons after each dump just before you use it again. And use plenty of water each time you flush, especially with #2. Don't ever skimp on water thinking you can extend the time between dumps. You are asking to get plugged up by skimping on water.
On the grey tank, as said above, it may be due to grease. Avoid putting any grease into the grey tank.
We have a Valterra clear back flush elbow. It kinda works okay, but I find that the tank rinser is better. The elbow is the next best thing to a rinser though. I don't use it for back-flusing anymore now that we have a tank rinser. I always use the clear elbow to see whats coming out and if it is clear water or not.
A tank rinser for the black tank really helps. We have a factory installed one and I can get the bottom of the tank looking clean as a whistle. This one is a Tornado Rotary Tank Rinser. If you have an enclosed underbelly, it will be a challenge to get access. And if you do pull off the coroplast material, I would upgrade to Horst Miracle probes at the same time. The standard OEM button sensors are next to useless.