The dropping of /t/ or /d/ in English is technically known as alveolar plosive elision. This phenomenon is completely different from the substitution of a classical /t/ with a glottal stop. In cases where we use a glottal stop, the stop can be considered an allophone, in other words an alternative form of /t/. In the case of elision, there is no substitute sound. The sound disappears altogether from the word.
Alveolar plosive elision
As a rule, when the sound /t/ or /d/ occurs at the end of a syllable (and a morpheme boundary), we can drop it whenever the following two conditions are met:
- It is surrounded by consonants (not including /r/ or /h/).
- The preceding consonant has the same voicing. (It must be unvoiced for /t/).
This means we can drop the /t/ in left work, because /f/ like /t/ is voiceless (there's no buzzing of the vocal folds). We can't drop the /t/ in halt work though, because the /l/ there is voiced.
This context will allow for /t/ or /d/ elision in nearly all cases in Gen Am and SSB English. However there are many other instances where /d/ or /t/ may be also be elided. For example, /t/ is freely omissible in normal speech in contractions with not - regardless of whether followed by a vowel:
- aɪ 'kɑ:n 'ɑ:nsə [I can't answer - Southern Standard British English]
- aɪ 'kæn 'ænsɚ [I can't answer - General American]
The Original Poster's question
The /t/ in first is a classic candidate for elision. It occurs at the end of a syllable, and indeed at the end of a word. It is surrounded by consonants and preceded by a voiceless consonant, /s/.
The /t/ in don't is also liable to deletion just because it happens to be part of negative contraction. Generally speaking, there are three main possibilities with regard to the t in don't. Generally, the most likely outcome is that the /t/ will be realised as a glottal stop. The second most likely is that the /t/ will be elided. The least likely is a canonical /t/. However, in this case, as pointed out in a helpful comment by Janus below, if the speaker is already using "wanna" there is a much greater chance of /t/-elision as opposed to a glottal stop in this example.
The Original Poster's transcription therefore represents a distinct possibility here.