Well, I have mostly run orienteering and trained for that, in Norway. With snow in the winter we were forced to the roads, but at any time else, we ran off road. Just get shoes that do not break apart (can endure being wet all the time) and continue your running. Who cares it is muddy or wet? If the alternative is to run on road and lower the milage, who would pick that?
There are plenty of off-road running shoes.
I did wash my shoes earlier, but I see no point in adding to the wear doing more than flushing off mud gently. Do NOT dry them in the sun or fast drying cause then the fabric might get stiff and shrink and damage the seems or glued parts. Of course do not hide them in a plastic bag, but let them rather air off for instance outside your door or in a place they do not bother others, away from direct sunshine.