If those trails never dry due to groundwater seeps or groundwater runoff, I agree with others who have suggested raising the roadbed. I think someone elso also commented that if groundwater is the problem and you put rock down without some kind of underlayment, be prepared to repeat it as the rock is swallowed up by the muck. I have a short stretch like that and I threw on all the loose (big) rocks I picked up around my place by the skidsteer bucketful. It took a more than I thought possible. I'm sorry for all the bad news, but that may be the reality.
If it's not groundwater, this works for about 40 inches of annual rainfall I get on my place. I cut all my trails half again wider than I need for one lane. I run on the upslope side when it's dry and the downslope side when it's wet. Coming or going doesn't matter. If I have to cross over, I do it in a higher (more dry) spot. The wheels nearest the middle of the trail are always between the other two tracks. The dry side is used more so it's mostly dirt, but I stay off of it when it's wet. The wet side gets torn up, but it's used infrequently enough that it dries some and mostly revegetates between uses. The vegetation really makes a difference on the wet side. Will fescue grow in your area? Otherwise maybe just a bunch of fertilizer to thicken whatever comes up natually. These may or may not work for you - I'm just trying to offer different ideas to consider. Good luck with whatever you do.