Hi J Wagon,
Usually the iPhone 5 and 6 is not as prone to sunk home buttons, as the iPhone 4 were, due to the way the home button is attached to the screen on the iPhone 5.
Replacing the rubber gasket will not do anything to the way the home button seats. The gasket is only there to ensure that not too much dirt gets into the phone through the home button.
Seeing as you say that the button has become difficult to press too, I would expect that the small metal spacer has fallen off - see picture below:
If ou look at the center of the button, there is a small circular metal spacer which needs to be there, for the button to function properly. If you have replaced the plastic part of the home button with a second-hand part, this might be missing or it might simply have fallen off.
If you do not want to but another button, you can try and make a small spacer yourself, of any material and gluing it to the back of the home button using superglue. This if of course not the ideal solution, but it is possible to make it work.