If the sign says keep your cart on the path, do so. Even if the sun is out. After heavy rain, the turf will be soft and the cart can damage it and maybe get stuck. Which is bad for the course and the cart. And it could cause an injury to you, too. It doesn’t matter that the course down the street is letting carts leave the path. No two courses are exactly alike even if they’re next-door neighbors.
You’ve heard the expression “sh*t flows downhill?” That really is true on a golf course. Silt, soil, grass clippings and other debris will head downhill when dislodged by heavy rain, and seek the lowest areas collecting there. Drainage and evaporation will be slow, so those areas should be avoided.
I heard of a club member who complained to the green chairman that “I noticed the fairway was really wet as I drove across it. When are we going to get the superintendent to fix this?” We’re good at our jobs, but we leave the weather to a higher power.