How come sometimes in Chess, I don’t have to check the king before a checkmate and then other times I do and it results in an annoying draw?

When playing against a computer, the computer will announce check first (as the King is threatened) before announcing checkmate (the player cannot neutralize threat to the King by any mean) immediately afterwards.

Most players will usually say "Check & Mate" or will reserve declaring checkmate as they have to review the opposing player's options to ensure they cannot get out of check before declaring mate (or checkmate).  If the game is timed, players may mutually stop the clock to access the state of the board as there's only 3 ways to get out of check.

1) Moving the King to a safe space, not threatened by an opponent's piece.

2) Capturing the piece threatening the King

3) Blocking the threat by moving another piece between the King & the opponent's piece.

In the cases of double-checks, Option 1 (Moving the King) is typically the only viable option since there's 2 pieces threatening the King at the same time.  Option 3 (blocking) isn't available since you're only able to block one threat.  Option 2 (capturing a threat) is rarely available since you would have to move the King to capture one threat & evade the other threat.  It's noted that it's impossible to get triple-checks or higher under standard rules as it would require a player to move into check (which is an illegal move).

IF the player can get out of check, they resume the game & make a valid move they discovered.  Otherwise, they confirm the checkmate & end the game.

YOU DO NOT HAVE TO PUT THE KING IN CHECK FOR ONE MOVE BEFORE GOING FOR THE CHECKMATE!  You can put the king in check & then confirm they're mated before ending the game.

The only time the game would end in a draw after a check is if the player moves their King out of check & the opposing player moves their pieces to prevent the King from moving, but not threatening their current position AND that player has no other pieces they can move (since players CANNOT pass their turn under standard rules).  Such actions occur when a player is inexperienced OR made an end-game blunder.

checkmate = king is in check and can't get out = win for the 'mater'

stalemate = king is not in check and no legal moves left= draw

the possibility of stalemate makes the game interesting and gives hope to a side that is losing otherwise eke out a draw, not to mention all the creativity in chess problems

As Nick has correctly replied, you are confused, & need a teacher.

If the king is not in check and has nowhere to move and the other pieces on the board cannot move then the player has no moves and it is called a draw. The goal is to trap the king and be able to deal the finishing blow. If they cannot move before you put them in the final checkmate then the player forced a draw, which is the best-case scenario for a losing player. Being able to force a draw is a high-level move that takes much practice. Just like a king and a bishop cannot force checkmate by themselves.

A way to look at this is by setting up the board with white having a king and a bishop and black only having their king. If black plays correctly they can force themselves into the corner and not be taken out by either white's king or bishop. This forces a draw and black does not lose.

You are confused. The king is always, by definition, in check during a checkmate.