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It sounds like you are having trouble with general nonogram solving techniques. Every puzzle in the game has been thoroughly tested (I would know!) to confirm that there is only ever one solution, and that the player never has to make guesses about where to mark. There are no secret solutions, there are always logical moves you can make to correctly solve a puzzle!

Here are some tips:

Don't guess! Look at the information available and mark tiles that you know must be filled or empty. Each bit of information builds up and gives you more clues as to your next move, until the puzzle is complete. If at any time you think you have to guess, you have missed something important. Guessing will introduce bad information into your solution and can make solving much harder.

Don't just mark squares you know must be filled, mark squares that you know cannot be filled with an X mark! Knowing where tiles cannot be filled is every bit as important as knowing where tiles can be filled. There are many puzzles where this information is the key to making progress.

Some more reading that might help:

How To Play Picross - Swadge

Nonogram Solution Techniques - Wikipedia

Make sure you check your album progress (Select button while in the overworld) as that is invaluable for tracking down what puzzles you are missing. All the puzzles in the first 100% are tracked in those albums. Those are the easiest puzzles to miss, and they do have a grouping scheme. If your albums are all filled out, you know to look in the endgame areas for your last two puzzles, and then the TaiFab+ puzzles should be behind that.

I'm glad you are enjoying the game! It turns out the Jellyfish isn't the only one in the game with an eye for rocks.

You will find someone else is involved in the gem trade if you look deeper in the Moon Cottage Bed and Breakfast.

I'm really glad you are enjoying the game!

If you are having trouble finding the Sea Angel, try looking in and around the Developer House north of the Museum

Good on you! I know losing progress hurts, especially after hitting the initial 100%, and it warms my heart to know you liked this game enough to overcome that setback and tackle the endgame puzzles!
Did you get the game running on a Game Boy emulator, or did you do another play-through in your web browser?

There are a number of Game Boy emulators that you can use, many of which are free and open source. While I personally use Bizhawk, something like Visual Boy Advance might be easier to get started with.

I love it, thanks for your hard work!
I would personally add two achievements for what I think are the most missed puzzles:
Stay hydrated (Complete the 8*8 Water Drop puzzle)
BEHIND the counter (Complete the 8*8 Skull puzzle)
I also personally would prefer [Annoyed burbling] for acquiring the Crab's Fish Power, but that's a matter of taste.
I personally found the 12 * 12 Grapes puzzle to be the hardest puzzle in the game, if there was an achievement for them specifically, I'd name it 'OF WRATH'
In any case, good job putting these in!

Are you playing in your browser? Depending on your settings, your browser may automatically clear your cookies, and in doing so delete your saved game. If you download the game ROM you can play in your Game Boy environment of choice, and have direct access to your saved games. Regardless, check your browser settings regarding temporary files and cookies if you intend on playing GB games in a web browser 

Why would this be needed? Even major Game Boy titles eventually dropped this as a standard, see Pokemon Red and Blue for example.
Additionally, TaiFab's splash and menu screens can be progressed with the A button in addition to the Start button, so having a hold at 'Press Start' seems a little misleading.
What would having a 'Press Start' on the splash screen accomplish. pray tell?