"I want to work in a garage."
Translation:Dw i eisiau gweithio mewn garej.
Not in this context. The pattern with eisiau is used all over Wales, although its pronunciation varies. The pattern with moyn is sometimes used particularly in south and south-west Wales.
Isn't there also a grammatical difference, in that eisiau is used without yn but moyn is used with yn? i.e. Dw i eisiau gweithio but Dw i'n moyn gweithio?
There is no difference in meaning between the patterns.
Yes, eisiau is a noun and the pattern dw i eisiau... is probably a simplification of the more formal but uncommon rydw i ag eisiau... where the ag has been dropped.
moyn is a contraction of the verb-noun ymofyn (which has some wider meanings than just 'wanting') and so yn/'n (or wedi in some tenses) is needed as a link to dw i, etc.