This was another type on the mixed round. It seems that “knapp daneben” almost works better for checkered fillominos than for the plain ones.
Rules Add or subtract 1 from each clue, then solve as a standard Checkered Fillomino.
To welcome the new year, here’s a Tapa “knapp-daneben”, i.e., all clues are off by one. The type features on Tapa Variations Contest XVII which takes place on the week-end. I’m really looking forward to that! For a more thorough preparation, check the practice puzzles on Bram’s blog. 
Rules Increase or decrease each digit by one, then solve as a regular Tapa. Digits may become zero, even multiple, but not negative.
One of the non-standard puzzle types in the Serbian Grand Prix round is Shikaku with clues that are off by one (as “Shikaku Liars”). Here’s a practice puzzle.
Rules Split the grid into rectangles composed of grid cells, containing one clue each, such that the size of each rectangle is equal to the corresponding clue ±1. Or see the instruction booklet.
There’s another easier puzzle of this type on the Rätselportal: Knapp daneben – Rechtecke. Then, in the knapp daneben instance of the Hausigel Contest Series. The double-clue Shikakus on croco should also be good practice: 2014-03-01, 2014-04-04.
Here’s a “knapp daneben” pyramid puzzle, i.e., all clues are off by one. Please excuse the slight asymmetry, it’s because the bottom clues including gaps were meant to be transferred from other puzzles. It’s fall-out of a puzzle contest I’ve helped write, together with the Berlin Stammtisch, which will be held on the weekend of June 28. See the instruction booklet.
Rules Place a number from 1 to 9 in each cell, such that for any two horizontal neighbours, the number between and above the two is their sum or their difference. In gray rows, all numbers must be distinct, while in white rows, there must be at least one pair of duplicate numbers. All given numbers are off by one.
Maybe Puzzles 