I wanted all my text to have a filled background to the size of the padding but because <p> and <h1> are block level elements it gives me some negative side effects, like the background being continued over the rest of the page.

I tried to change the display to inline and it works, but then i lose the pagebreaks and formatting I would like to keep with the p and h1 tags. Does anyone know of a good way to keep the block formatting and spacing, but while being able to color the text background so that it "hugs" the text and not stretches over the whole page?

block elements are 100% wide by default so they are not negative "side effect" just the correct behaviour.

If you want inline behaviour from block elements then set them to display inline (as you have) then add other elements to control the formatting.
eg:
If you want a line break above and below a display:inline heading, then simply add one in the source code.

Alternatively, set fixed widths on the block elements.