Some assumptions are being made with my response. The water leaking and ice damming are sort of related, but need to be understood separately.

Water Leak: Your manufacturer's asphalt shingles are more than likely able to be installed on a 2:12 slope (the minimum). However, the underlayment requires specific installation in overlap or multiple layers, or using an ice and watershield under the entirety. That may not have been done.

Ice Damming: That process can cause water to back up under asphalt shingles regardless of roof slope. But even if you repair the shingle underlayment properly for 2:12, you still don't want ice damming as water finds a way. And, fixing the cause has other benefits.

Here it appears the cause is because you have an enclosed rafter assembly on a low ceiling. You would not have enough room for venting and/or enough R value of insulation, where in a snowy climate probably requires R60. So the conditioned room (warm) meets the roof sheathing (cold), causes enough melt, refreezes, cycles over again, and eventually backs up under shingles.

Redo the roof pitch: That is a choice. Besides topping off the roof, it would require new walls to match the original house's top plate height (but would allow normal height windows and ceiling). But I would question if the foundation was done correctly, this type of addition is reminiscent of a DIY buildout of a former concrete slab patio.