Is dwi ‘I’ and dwi’'n ‘I am’?
Not quite, but you’re on the right lines. 
“Dwi” is an informal way of writing “Dw i”.
“i” is “I”.
“Dw” is like “am”, but Welsh works differently to English, so you’ll need to say “Dw i” every time you want to talk about yourself in the present tense (whereas in English you can pick between “I play” and “I am playing”, you’ll always need “Dw i” in Welsh) .
As for the “yn” I wouldn’t worry about it for now. Just try to use it when you hear it.
They are both “I am” because ‘dwi’ is the 1st person singular of the verb ‘bod’ (to be). The 'n (yn) is added when the next word is a verbnoun, noun or an adjective.