Do you want the eaves of the new addition to be higher than your existing eaves? If not, I don't see how the height of the existing eaves factors in. The new addition with a shed roof will give the new room the same ceiling height as the existing room where the sliding glass door is.not sure we have enough height to do it that way the eves are only about
6' 8" above the deck! Oh and this doesn't get much snow it's on the Washington Coast!
I agree that anything other than a new roof will be a compromise and will look patched together. While not ideal, a shed roof extended over the new addition would give the OP a 2/12 pitch, maybe 2.5/12 pitch. In our area here in the south, a lot of double-wides have 2/12 pitch roofs without leakage issues. That shallow of a pitch would be more of an issue in a place that gets lots of snow. The shed roof over our back porch is a 3/12 pitch and is no problem at all.I would second the new roof concept, getting trusses premade would make it a big, but not gargantuan job. You would have a much better performing, looking, place when you got done. The difficulty is that the current roof peak elevation isn't enough to get you good drainage all the way out onto the new addition.
A separate place over a new garage kills two birds with one stone, though!
Chris