If they don't sign him, wouldn't he become a free agent?
or can you trade an unsigned player?
Nope, I believe he goes back into next years draft.
There's a deadline by which time the player needs to "re-enter" the draft, but the teams do not hold their rights longer than a year or so, you are correct.
The most famous example is probably Bo Jackson, who was drafted 1st overall by Tampa Bay, but chose instead to play baseball for Kansas City.
He went back into the draft the next year and the Raiders took him in later rounds (the 7th, maybe?) and did that deal where he could keep playing baseball while in season and then play football later once the Royals were done.