When ever I prep a deck for restaining, I clean it with a mixture of TSP, bleach and water. I apply the mixture, scrub it with a stiff broom and rinse it well with water or a power washer. Let it dry well, then apply stain. The cleaning solution helps remove the dirt, grime, spills, etc. Pretty basic stuff, I admit.
As far as why your stain doesn't want to dry. There may be several reasons for this. First thing that comes to mind is that the stain may have not been mixed well before application. The solvents and drying agents can separate from the oils and pigments in stain. The oil tends to stay at the top of the can, and the oil won't dry well without the solvents mixed in.
Another reason may be other sealers, grease or spills that have sealed the skin of the wood, not allowing the stain to penetrate.
Sometimes, one can get a little carried away and apply way too much product. This will saturate the wood and the access will sit on the top of the surface.
If after a couple weeks of good drying weather, you still have sticky or tacky spots, I'd be tempted to remove this excess with a cloth wetted with mineral spirits. This should thin the sticky stain, remove some and distribute it to help it dry.
For the main deck, I would recommend a floor sander, similar to what's used for finishing hardwood floors. Put a 100-120-grit belt on it and take it back and forth along the deck in the same direction as the planks until you see bare wood. Once you have bare wood, I would apply whatever stain or sealant you want very quickly thereafter, as you have just removed a large portion of both the artificial and natural weather and bug resistance the wood had.
A handheld orbital or belt sander will get most of the stuff off the vertical components (handrails and pickets). You'll need a "corner cat" or other detail sander, or some elbow grease, to get the paint out of crevices and detail work.
You will not be able to avoid taking off some wood along with the paint. Paint (especially a product designed for this kind of outdoor application) will seep into the grain of the wood, and you will be removing wood before you see the color disappear and bare wood show through.
By the way, you will still need to maintain the deck after staining and sealing. The stain will last for quite a while, but you'll need to re-apply the water seal at least once a year, once every 6 months if you get a lot of rain.
Best Answer
Rent a drum sander and run it in line with the boards. You can take off a large amount of wood in short order depending on the grit of paper you use. Then use a sanding disc with a backer on a grinder for detail work where the drum will not go. We used 36 grit to bring some weathered wood back; probably took of over 1/4" off the top in that case! Total out of pocket should be around $200-$250 for everything.