I don't think offside rule itself is the problem nor fix. It should be tweaked, but in my remedies for the Game offside rule may not even be necessary.

Fubdamental Problem: it is so pathologically difficult to score in the current Game (last nights blow out not withstanding) that it too often leads to the anomaly of an inferior team getting a result against a superior team. This difficulty also results in an unhealthily weak correlation between superior play and scoring in a typical game. The reason for this fundamental problem is due to the current structure of the Game related to size of field, number of players, and pace and space given those first two things. The Game is largely in an archaic state, having changed little in terms of structure since the late 1800's.

Fix: Reduce the length of the playing field (keep current width), reduce number of players (I recommend 7 field players). This will result in more frequent chances on goal, more goals, and probably improve the correlation between superior play and scoring.

Also: fix some of the current rule stupidity. Such as fouls in the box necessitating penalty kicks thereby creating a ridiculous offence/punishment imbalance (my recommendation is awarding direct kicks for most box fouls). Clear up some of the typical ambiguity and lack of definition in FIFA's Laws of the Game.