Learn Dutch: Ik ben gewoon moe.

You are right! As an adjective, gewoon means "normal" or "usual". However, Dutch speakers frequently use it as an adverb to mean "simply" or "just". In this sentence, it emphasizes that there is no complicated reason—the speaker is simply tired.

Can I use net or maar instead of gewoon to translate "just"?

Not in this sentence. The English word "just" has a few different Dutch translations depending on the context. You use net for time (as in "I just woke up") and maar to mean "only" (as in "I have just one euro"). Because you mean "simply" here, gewoon is the correct choice.

If I wanted to say "I am just very tired," where would I put the word for very (heel)?

You would place it right before the word it describes (moe), resulting in: Ik ben gewoon heel moe. The word gewoon generally stays near the verb, while heel needs to stick directly to the adjective.