We've had lots of problems with swampy ground where I live. Soil is very sandy and it creates a slurry. The surface grasses hold up the tractor sometimes, but then the front tires sometimes cut through the surface grasses and create a rut that back tires fall into and widen and then you're stuck good. Just have to keep an eye on those front tires as you go along to make sure they don't start sinking. Also never travel the same path twice, use fresh ground every time you go across soft areas. Use the loader to push backwards if you get stuck. I also find that driving the tractor backwards through swampy areas seems to work better. When going backwards, I think the larger rear tires pull instead of push.
Typically, there will be dry spots or raised spots that are solid, and lower wet spots that are not. So just mow what you can, and leave the rest. Any mowing will help. If you have hay equipment, maybe you can mow and rake the edge of the field and repeat and advance a bit every two weeks as things dry out.
Overall, the best solution we've found for swampy soil is to create a way to drain out the water. Make raised pathways through the fields during the summer so you can get across your fields in the winter. Works great if you can add gravel or a plastic paving fabric to cap the ground so water runs off instead of infiltrating the soil. To go with your pathways, add the drainage ditches which should be very deep if possible. At least 3-6 feet, and 3-6 feet wide (use this removed soil to build up your paths). Capilary action in the soil will draw water out if there is a ditch for it to drain into. So the challenge is digging during the summer with winter in mind. Take pictures to remind yourself where the really bad flooding occurs and deal with those areas during the summer. It may look a bit ugly the first year, but once its grown over a bit, it won't show.
Of course, if its a hay field, its nice to have it be level and flat. But you can still crown the middle of the field with a road/path and add drainage ditches around the edges. Then you can still cross the field in the winter, and you can mow the edges of your new road.
We've laid many rolls of 4" flexible perferated pipe as well. However, it gets clogged fairly quickly with sand. We tried putting gravel around the pipe, but still the dirt/sand will clog the pipe within 2 or 3 years. Water still wicks through, but not as well as when you built it. The trick is to use some sort of filter fabric or landscape fabric down first in your trench, then put in some gravel and then the pipe, that will stay working for a while. The fabric encasing the gravel and pipe will hold back the dirt/sand from infiltrating the gravel. The air spaces in the gravel are what gives you suction underground to draw water out of the soil and into the perforated pipe (plus the airspace in the pipe itself).
Oh, and you don't actually have to get the pipe level. It can rise and fall a bit and will still carry water, even if the end is the same height as the beginning. Its not gravity that drives the water out of the soil, its the capilary action and airspaces in the pipe. So don't get hung up on leveling or sloping the pipe.
Landscape fabric can be expensive (like $25 for 3 foot by 50 linear feet) but I found that filter fabric can be purchased in 12-foot wide rolls that have 400 or 500 linear feet for approx $300. Available at more professional building supply places (not Lowes). Thats the way to go.