According to my Japanese-Japanese dictionary (published by Obunsha), "たくさん" is not categorised only in なadjective, but also in noun and adverb. So, "たくさん" in "たくさんのしゅみ" is noun, and "たくさん" in "しゅみをたくさん持っています" is adverb. If we say "もうたくさんだ", then it change into "なadjective". Very confusing...

Yes, しゅみをたくさん持っています is absolutely correct. たくさん is not used as a na-adjective when you want to mean "a lot of" things. It can be used as a na-adjective when you want to say something is "filled with something". For example: 具がたくさんなスープ (soup filled with ingredients) 自然がたくさんな街 (a town filled with greenery) However, this usage is not very common and you can still use たくさんの there instead.

When people say たくさん of something, they are saying that there are a great amount of something. So it's usually something we can count or measure. I can't explain why you wouldn't say "たくさんな", but I can say that it's always "たくさんの" when modifying a noun and "たくさん" when modifying a verb. たくさんのしゅみを持っています。 しゅみをたくさん持っています。