wire nut alternative
You are possibly using the wrong size wirenut for the number and size of wires you are connecting, or simply installing them incorrectly.

"Saving money" is a dubious justification for tackling electrical work if you are not up to the task - the money you "save" can easily cost you a good deal more when your work fails in a more dramatic fashion and burns the house down or injures/kills someone. You could hire an electrician to go over your self-installed wiring and make sure it was correct, and odds are excellent that no more wirenuts would fall off after that.

There are "push-in" wire connector blocks made and listed now. I'm not personally a fan of them, but they are made and listed.

i.e. See this image from Ideal Industries (not endorsing or associated, just using the image as an example)