Yes it is an exception to the rule. As you say, when an adjective is used as an adverb you normally add the -t ending. There are however exceptions, especially for adjectives ending in -ig, -lig, and -vis. Some of these adverbs will always end with a -t (for instance 'hurtigt') and some of these can end either with a -t or without a -t (for instance 'jeg vågner tidlig(t) om morgenen'). Sometimes the meaning of the sentence will change depending on the -t
So this is where it gets weird, because 'stadig' as an adverb can never end with a -t thereby contradicting the rule just mentioned. Then I looked 'stadig' up, and it appears in the Danish grammar as a 'stand-alone' adverb having no connection with stadig as an adjective, thereby leaving out the -t ending. So I'm a little puzzled actually...
Further explanation here: http://sproget.dk/raad-og-regler/typiske-problemer/adverbielt-t/adverbielt_t_uddybning . There are also some lists of adjectives and how they appear as adverbs
I don't think it's an exception as such. "still" just means "stadig" in Danish, and it doesn't really have a lot to do with the adjective "stadig".
Otherwise all other adverbs would also have to end with a "t", for example "morgen, nok, i dag, måske, .....)
I think in most of the cases, only adverbs that are directly related to a certain adjective end with a "t" (though there might be some exceptions, too).