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.
Best Answer
Hard to tell whether you have screws or nails holding everything together right now. It is either nails - which can, relatively easily, come apart, or screws which can come apart but not as easily.
If you have nails then I would reinforce using some long (e.g., 3") screws and you should be set. I would want two screws into the end of each tread. The railing is sometimes a bit trickier, but screws to supplement the existing nails is a good start. If that doesn't take care of all the wobbling of the railing then you may need to replace parts or add some additional supports. Of course, if you find any cracked or rotted pieces, or if anything cracks when you drive in screws, then you need to replace those pieces.
If you already have screws then I'd be a bit more concerned as that may be a sign of the wood rotting (and therefore not holding the screws any more) or that there is just too much movement of the ground causing everything to come apart.