Trees pretty much keep the rain on the trails at a minimum. No trails get closed that I know of. Most shoes r ok !!
Also some of the trails, you have to walk down the side of a ridge. It can get real slippery. There are dirt trails that get pretty muddy and there are gravel and asphalt trails. I'd stick to the latter two after it rains. Trails are only closed when they are completely obstructed, washed out, or too dangerous. The most popular trail is along the creek, and it has been washed out before and reconstructed. Regular hiking shoes are okay. I've hiked in hiking boots, sneakers, and slip on shoes, depending on the weather and trail. Use your best judgement.
Lot of variables...depends on the trail. Not sure that Nat'l Forests close trails for mud unless there's a real danger. True hiking shoes of decent quality should handle trail wear including normal mud. Probably still gonna get wet inside. Call the park HQ on the day you plan to hike for trail conditions/closures. Some muddy areas are probably avoidable.
Regular hiking shoes should be fine but it can get muddy - not too muddy, though.
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