"My name is Adam."
Translation:Nazywam się Adam.
No one speaks like that.
Either "Mam na imię..." (for given names), "Mam na nazwisko..." (for family names), or "Nazywam się..." (for either or both).
The above is only for humans, for animals you can use either "ma na imię" or "wabi się", and for objects you can only use "nazywa się".
Better say "Moje imię to Adam". It is maybe a bit unusual, but not awkward at all. The particle "to" often replaces the verb "być" (to be) in Polish, which is easier, because you may follow with the noun in Nominative. You would say for example "Warszawa to stolica Polski" as well as "Warszawa jest stolicą Polski" (Warsaw is the capital of Poland) or "Marek to dobry uczeń" rather than "Marek jest dobrym uczniem" (Marek is a good student).
"Adam mi na imię" or "Adam się nazywam" or "Na imię mi Adam" or "Imię moje to Adam" are also correct and really not so awkward or uncommon.
Well they all sound like playing with language, and at this stage of learning maybe it's too early to encourage the learners to do so ;) "Na imię mi Adam" seems most natural to me from those options and I added it.