I'm fortunate I can park my camper on asphalt at home. But I do occasionally park it on the grass. When I do, I always put boards under the TT tires, 8 footers. The tires will not sink into the ground, and this will enable the trailer to start rolling with a bit of momentum before hitting the grass. Once moving, it keeps moving.
To get your camper moved now, you may just have to wait till the ground gets good and hard and dry. You can do a number of things to get more traction on your tow vehicle. Does your truck have 4x4? If so, engage 4x4 and it will probably pull right out. If not, you may just have to wait, or call a tow truck with a good wench.