Yes, "gotta" is "(have) got to", whereas "(be) going to" comes out as "gonna". Don't ever put these in your writing!! These are just how we express a mumbled form of regular grammar. Even when speaking, if your stress and rhythm is not entirely natural, you could sound childish. Stick with normal, regular grammar and real words, and you'll have no problem.

Gotta is the short version of 'have to' or 'have got to' and it is used more in the USA rather than the UK.

Gotta is a slang that means GOT TO :) I gotta go the hospital = I got to go to the hospital .