Creating a Puzzle Game (Tetris Style)

I want to create a puzzle game with the basis of tetris, where there are different style blocks (L, T, etc) and you have to fit them all together. However I want to make it a little different where there is a set-size grid and once you fit all the pieces in the grid you move on to the next level.

Each level will have a set of blocks that don't change, and you must use all of them to finish the puzzle.

The way I have it now, the grid is made up of a double array of Game Objects that detect when the block is placed on them, and snaps to position. This is very very unstable and messy.

Is there a better way to handle grid placement like this? Also, is there a way to detect when the map is completed and the player can move on?

I don't need specific scripts, I just want to know if someone can point me in the right direction to make a grid with snap-able tiles.

For Reference, look up 1010!

Thank you everybody! (Trying to post this before my computer dies since I'm at 2% so sorry for any mistakes in the posting)