Translation:I would move to America, if I were able to.
The byddwn i, byddet ti, ... forms are used as an alternative to the baswn i, baset ti, ... forms sometimes, that is all.
From a point of view of style, though, if you use the taswn i forms in one part of the sentence, it is probably better to stick with the baswn i forms in the other. Grammatically, it makes no difference in the colloquial language.
Could I think of the 'byddwn i/tasiwn i thing as 'I would if I could? Would that be a fair way to remember it?
The "if I could" is specifically "taswn i'n gallu".
Just "taswn" is "if" (for a more hypothetical / unlikely / counterfactual event) or perhaps "if I were" (and then taswn i'n gallu would be "if I were able to").
taswn i means 'if I were to be'.
The 'if' in question here is in fact the word pe/ped which, when added to baswn ('I would be') leads to the form taswn:
- pe(d) + baswn i -> petaswn i -> taswn i
pe is not often heard in the colloquial language because it tends to get rolled into taswn i, taset ti, tasai hi, etc, but the form pe bai (as an alternative to tasai) is quite often used in spoken Welsh with the meaning 'if it were to be'. It is the normal form in more formal and written Welsh.