Currently your bike has tires with a smooth tread appropriate for street.

They will work off road in dry conditions adequately but, as you have found, not so good for mud or wet grass.
Tire Size
You can purchase tires sized 20x1.75 with varying levels of knobby aggressiveness. If you keep your fenders you will be limited to tires with the same width as your current tires. Removing the fenders might (but only maybe) offer a little more room for a wider tire. Your frame will be the limiting factor. Wider tires do help off road.
Tire Aggressiveness
The following tires were offered in 20x1.75
This tire is something between an aggressive knobby and a street tire.

A more aggressive tread would look like this

This tire will generate a buzzing sound on pavement.
It's all about compromises, smoother tires roll better on the street. Knobby tires offer better traction off road.
Tire Pressure
Tire pressure does matter. This is also all about compromises. Lower tire pressure offers a better ride and better traction. If the pressure is too low you can pinch the tube and get flats on bumps. Higher pressure generally rolls better on the street. Too high will blow the tire off the rim. This takes some experimentation to find the right pressure. "The right pressure" will vary with the situation.
Tire Width
There are times when having a different width tire on the front or back makes a difference. Usually, the difference is small and in your case, if you are keeping the fenders, you don't have a choice.
Tubeless on Non-tubeless
You can put tubeless tires on non-tubeless rims by using a kit to convert your rims to tubeless compatible. Knobby tires will give you the most off road improvement in traction. I don't think you'd notice any difference going tubeless.
Selecting Your Path
Wet grass and mud are tricky. In some situations it's just plain slippery even with aggressive tires. Paying attention to your path and choosing the correct trail can be more important than the tire you have. If your path has a rut, the middle of the path often collects water and is muddy. Ride on the edge of the path, drive your tires where there will be maximum traction for the path you are on.
Conclusion
It's going to take some experimentation time and money for tires to improve your wet grass / mud traction issue and in the end you will compromise the street riding aspects of your bike.
If selecting a different route - one that is all paved - is an option it might be a better way to go.