There is no reason to put it down. One thing I've noticed about spiders is that the wobble gets worse when they're excited or stressed. It's possible that the wobble will become less noticeable once your snake becomes more used to it's new surroundings and becomes more comfortable. Yes a 20 gallon long should be suitable for your snake for a very long time. It won't matter how strong the snake gets, it's wobble will not break a glass tank. Make sure it has a secure escape proof top and cover most of the top with a piece of clear plastic to hold in humidity. when checking your temps, make sure you measure it on the floor of the cage where the snake is and not half way up the side of the cage where it will be much cooler and the snake will not be spending any of it's time.
The wobble is the reason I prefer not to work with the spider mutation anymore. There are various other morphs that are also prone to this like sables and champagnes but not usually to the extent that spiders are. Somewhere on this site is a thread that points out the various problems with certain mutations but I can't remember where it is right now, maybe someone else can post a link.
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