This is one of the problems with front wheel drive cars, I've found, and it's one of the reasons I wouldn't have one until it became near impossible to find a decent rear wheel drive one that suited my needs. I now have 4WD.
The weight on the tow-bar presses down on the drive wheels on RWD drive cars, and tends to lift the front on FWD.
I've only ever towed with a front wheel drive vehicle but there is a secret solution to towing off wet grass with a single axle van unless it's absolutely water logged and that's when a motor mover comes into full use. The solution is to tow the caravan off at an angle so the one wheel remains almost stationary when moving off so only one caravan wheel is in motion. Once you get moving the idea is to keep moving and NOT to stop but to stay in low gear till you hit solid ground. This method works on the same principal as a vehicle with caterpillar tracks such as a bulldozer or military tank but if you don't believe me just try it for yourself and see. I was given this tip a few years ago and not failed yet.
Bear in mind that if you are losing traction on grass you will also be losing it on the road although it won't be as noticeable.
Are you sure that your nose weight is not too heavy?
DaveS1