This is not an "expert" at work - just following through the prescriptive language of the IRC like a carpenter.
Working from IRC table for: Roof live load = 20 psf, ceiling not attached to rafters, L/Δ = 180), 10 psf dead load
Spruce-pine-fir #2 16" on center (????) 2x6 will span 14-4 (span is between inside edge of top pate and ridge - so maybe its really 11'-8"?)
If you raise the ceiling joist (what the code terms the "tie" at lower end of rafters) 1/3, the span is reduced 1/3 - so the 2 x 6 would only be good for 9'-7", not 11'-8"
The adjustment for raising it 1/5 - 14.4" - is .83 so 2x6 with ceiling joist raised 14.4 " will span 11' - 10 3/4" - so rounding off you could probably raise it 15" and still meet code - assuming all my assumptions on lumber grade and species, on center spacing, etc., are correct.
The allowable span of the same material as a ceiling joist is 16'-11", and raised 15" I think the span would be around 18' with no (storage) load - so a little undersized - just keep some hangers and the strong back, or change to 2x8s.
I don't think I'd raise them the 1/3 if it were my garage but probably wouldn't be afraid to let my kid into your garage in a snow storm if you did. Working with overhead loads and rigging, whether or not you'd allow your children under them is kind of the ultimate measure.
I'm happy to make run this again if my assumptions are off.
Was interesting - doug fir #2 would pass; hem fir and southern yellow pine are very close - like an inch or two off the 11'-8" mark with joists raised 24".