Do I need the hyphen?
I would. Hyphenation of compound adjectives can be a style question as much as grammar. Some style guides prescribe specific words that don't need hyphens, or can be combined as one word. Check AP or another guide for examples.
In both sentences, the noun is separated from the compound adjective (by "regular" in the second case). If you believe in hyphenating part-time and full-time as adjectives, the separation doesn't change the grammatical function.