It makes a difference how your garden beds are arranged, but I imagine that I would rake and lift the soggy old straw out onto the paths as path mulch, and get fresh crisp/clean straw to mulch the strawberries with. The straw will get trampled into the mud and decompose quickly (and invite worms), then later, I would scrape up the good topsoil from the path for rebuilding the veg garden beds and planting, for hilling, etc. before mulching the paths with other less easy to decompose materials for the summer.
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