Why not just buy the 375 JDJ now, and work up loads from small to tall as you get used to it?
It's really not as bad as people make it out to be. My 14" 45-70 muzzle tamer barrel kicks harder (with smokeless loads) than my 16" braked 375 JDJ does.
You could start with the "starting loads" and the 220 gr flat point bullets, and I bet you'd be surprised at how mild it really is. Work your way up from there.
If that seems like too much of a good thing, then how about a 309 JDJ instead?
I have a 14" 35 Rem bbl, and I would personally not consider it very good elk medicine. Sure it will kill an elk, under ideal conditions, but I think there are too many variables you cannot control when hunting to risk it. I like having more of a power margin. Elk are a majestic animal, and deserve to be killed with a swift and final end.
If you were in the Phoenix, AZ area, I'd let you shoot mine to try it out.
The best solution would be a 375 JDJ for elk, a 309 JDJ for big deer, a 35 Rem for lesser big deer, a 30-30AI for medium deer, and, well you get the idea...