The adjective for 'shit' is 'shitty'. Is there no different adjective for the British word 'shite'?
The adjective form of shite is shite:
That was a shite film.
I feel shite about it.
Mondays are always a bit shite.
Shite has an adjective sense, so for most uses, shite would be the adjective.
However, this is also true of shit.
Shitty overlaps in senses with the adjective senses of shit and shite, and often if you call something shitty you could also call it shit or shite. However, shitty also has the sense "covered in shit" which shit and shite do not mean.
In some dialects there is the adjective shitey which has this meaning. It would be the Scots equivalent to shitty and hence more likely to be found in Scotland and Ulster due to the Scots and Ulster-Scots influence, though it will also be found in other places, such as some parts of Ireland without as strong an Ulster-Scots influence.
Hence in such dialects the adjective you seek is shitey, as in the Scots expression, "he'd drink through a shitey cloot" ("he [was so-enamoured with alcoholic beverages that if it was the only way to get a drink he] would drink through through a shit-covered rag").
Since shitey is not as wide-spread as shite, and since shite is just a variant of shit, in other dialects the adjective would be shitty.