Hi, as you said, たくさん is adverb, so it's used with "vreb" though we sometimes omitt the verb in coloquial. (However, たくさんの is adjective.) About 多い, it's adjective and used to express the situation. Plus we also use 多くの for "noun". For example, "there are many questions" is translated in several ways. しつもんがたくさんあります。(adv.+ verb) たくさんのしつもんがあります。(adj.+ noun) しつもんが多いです。(adj. for the situation) 多くのしつもんがあります。(adj.+ noun) I think one of the best ways is repeating using them and being accustomed to them. Please keep up. I hope this helps you.

人沢山いるね。 人が多いね。 Those are the two examples that my boyfriend gave me when I asked for clarification from him XD It probably doesn't help all that much but I can say that you can say "多すぎ” but you can't say "沢山すぎ” not that people wouldn't understand but he said it just wouldn't quite sound natural.

子供はたくさん集めていますね There's alot of children gathering around here isn't it? 人は多すぎます There's too many people. I suspect that these two examples will differentiate the two for you.