You should probably remove excess stain before it turns tacky. Remember, stains seep into the wood, they provide no protection being on the surface.

You want the surface absolutely dry before clear coating. If the stain doesn’t dry off, I would suggest applying some solvent- whatever is recommended for thinning by the product to remove the tacky stuff if you were trying to recover it.

For sanding complex shapes, you might be able to get a tumbler or vibration tank. Sand blasting is very aggressive, even using finer grits; and plastic is very soft. I’ve had good luck with spongey dremel wheels, if one’s in a hurry, or, using foam sanding pads.

What did you use to stain it? It's possible you applied it too thick and it isn't going to ever dry. If you clearcoat over tacky stain, you're just going to create a bigger mess. You can let it sit longer and sand if off, or use a solvent and try to remove the excess. But the more you mess with it, the more of a mess it is going to be.

Might need to make some smaller test prints and experiment with finishing.

How long did you leave the stain on before wiping it?

What type of stain did you use? I imagine it makes a difference considering the part is plastic.

I’ve worked with a lot of Wood PLA (almost exclusively Amolen). It stains fine, but I always apply a very small amount to a fine bristle brush and use that to apply it to the parts. I don’t use much, and then I let it air dry in a room that’s at least 70° overnight before doing anything else with it.

I don’t know that I’d ever sand blast. I’ve manually sanded (which has allowed me to better fake “grain” before applying stain.