I'm kind of torn on what to recommend here. I don't like saying, "Go buy a different bike" all the time, though I share your concern about the gearing. If you've currently got a 30 tooth small ring and a 34 tooth cassette, I think you're looking at a 25 gear inch low. If you can find a (much) bigger 8 speed cassette, and appropriate derailer and/or adapter, you might get two lower gears (down around 21 gear inches, similar to the Disc Trucker), which will really help with the steep stuff.
OTOH, there's a good chance you'll still find hills that are so long, and steep, and you've been riding so long that you're tired, so you get off and walk anyway. There's just two gears' fewer hills to walk with the stock bike.
The other thing I'd be concerned about is hauling luggage. Are you going to be using full on bikepacking gear, or a more traditional rack and panniers setup? How will the bikepacking bags affect the bottle cages, if you go that way? If you go for rack and panniers, can you attach them to the Diverge? It looks like the Diverge's carbon fork is drilled for fork leg bags, but those will be smaller than full-on front panniers. And while you can attach a rear rack to the dropout (I think, based on peering at the "enlarged" picture on the web site), those slightly goofy chainstays may make a P-clamp attachment to the rack top ... interesting.
So just maybe getting a Disk Trucker would be the better way to go. Order now and hope for March delivery. And ride the heck out of the Specialized until it gets there. More saddle time in training is always a better thing.