1) SNGDLs are the real deal!
2) It is always amazing how out of shape the "in shape" people are..... I do GTG work frequently, it works. Not so much on high rep stuff, but gymnastic strength stuff, etc.
3) You'll need to do some specific work on your press. You might try some light warmup presses as part of your warmup, then do some low volume press work prior to the WOD. 2X5 or 3X3 will be plenty at your stage. Keep the volume low, and do the press work around twice a week.