Is shittly a word (and if so, how many t’s does it have?) or do I have to use shittily?
Generally, to make an adverb from an adjective like shitty, you change the -y to an -i and add -ly.
E.g.: prettily, cattily (bitchily), dottily (eccentric), grittily, knobbily, shabbily, etc.
Yes, it is a word.
shittily (ˈʃɪtɪlɪ) adverb (informal, rude): in a bad way; very badly
FYI: The word shit has a long and well-documented history. Anglo-Saxon leechdom books use scittan in reference to cattle having diarrhea. A Latin text from 1118 refers to "Lues animalium, quæ Anglice Scitta vocatur, Latine autem fluxus interaneorum dici potest." *
There are many examples of the verb from the 14th century [e.g., from 1387: "þey wolde ... make hem a pitte ... whan þey wolde schite ...; and whanne þey hadde i-schete þey wolde fille þe pitte agen." translated into ME: "they would... make them(selves) a pit...when (? where) they would shit; and when they had stat, they would fill the pit again."]
The noun is attested from the 16th century, both in reference to excrement and to contemptible people.
*roughly: Pay of animals, which in English is called Scitta, in Latin, however, one can say "the flow of her insides".