Learn Finnish: Minä olen nälkäinen ja väsynyt.

Just like "and" in English, ja simply links two words of the same type together. Here, it smoothly connects the two adjectives nälkäinen (hungry) and väsynyt (tired) without causing any tricky word order changes.

I previously learned to say minulla on nälkä (I have hunger) to mean "I am hungry". Why is it different here?

You could say minulla on nälkä ja olen väsynyt (I have hunger and I am tired), but it's a bit clunky because you have to switch from "I have" to "I am". Using the adjective nälkäinen (hungry) instead of the noun nälkä (hunger) lets you easily attach both feelings to the single verb olen (I am).

Is it okay to drop the word minä?

Yes! You will very often hear people just say olen nälkäinen ja väsynyt. Because the verb form olen already specifically means "I am", the pronoun minä is completely optional.

Is väsynyt a normal adjective?

It acts exactly like one in this sentence! It actually comes from the verb väsyä (to get tired), and the -nyt ending is a bit like adding "-ed" to a word in English. At this level, it is easiest to just memorize väsynyt as a regular adjective meaning "tired".