It is quite possible that you received a bad head from Evans, I've had a few which I returned to the Five Star shop where I bought them. Mine were G2 coated and clear heads and from the feedback from the guys at the shop, mine were not the only returns.
I would take the G1 off the drum and re-seat and tune it. If you don't have better results, return it for something different, maybe a Remo Ambassador, lol.
Sounds like a great shop. I've only been able to return heads (all with coating issues) by calling/emailing Evans directly. Same with Aquarian. Aquarian was much more responsive (i.e. ROY BURNS!) although both did follow through. Returning a head to a brick and mortar or online store because it doesn't sound good might be a hard one if there isn't a very good relationship in place already.
I don't know if it matches your type of music, but on most of my snare drums I use an Evans Power Center Reverse Dot batter head.
I currently have a G1, G2 and Power Center Reverse Dot. all coated, on 3 different snares (Remo CS Underside Dot and Aquarian Modern Vintage on another). I bought the G1 because I'm working through some brush studies off of the Brush Secret DVD as well as the Ed Soph Evans YT videos and wanted to go for a similar sound palette. The G2 has always been a winner on my Pork Pie 5 x 13 which seems to favor this head for the sound I want. The Power Center I've used before with mixed results sound wise but it's sounding very nice after a few days break in period.
The G1 sounds good on my Acrolite but - not a love thing yet. The coating is holding up better than the Ambassador that was last on there (I play a bit of brushes on it) but it's not getting the sound I want. I'm going to return to a single ply Aquarian choice (Either Satin TC or Studio-X) for the next head. In the meantime, it's not bad and is playable for the majority of the situations I'm using it for.
G1's have been the more challenging head for me but I can still get it to sound good.
The OP mentions being scared to mess with his reso. Might I suggest going one step further and replacing the reso and start experimenting? We as drummers need to go through this process many times so we can get this skill under our hands and in our ears. Lots of time it's trial and error. It seems like a small investment to be able to get head changes consistent through our playing career.
BTW - if that Amb had a 'dimple' in the middle, it was wasted way past when it should have been changed. I've never played a drum with a head in that condition that I felt could be used professionally for my work and I'm not the hottest pro on the block. Also I'm assuming that the resonant head is pretty old and worn out as well.
I say put some time into what you have in hand. Then replace the reso if it's still not working. Tune the drum all the way to finger tension and then carefully bring both up in pitch together until it starts to sound good. Then you can start to experiment with batter vs resonant tensions and tuning range 'sweet spots'. At the very least, change the snare side resonant at the same time you change the batter next time.
My .02
Jim