You shouldn't have had to assign the font-family to all p elements in order to get it to work for just the h[123] elements. It looks like you did everything right (so long as you actually added the ttf font files to the epub), so just remember that you don't really want to use Sigil's viewer as the sole means of verifying that your embedded fonts are "working" or not. Use ADE or an actual device to know for sure.
P.S. Not sure why adding the Georgia font-family to your paragraph style would make your Garamond fonts start working.
