I am assuming you have a contender carbine and not an Encore, since you mentioned those two cartridges.
I have a contender and use the .375jdj in the rifle configuration and recoil is stout, but not that bad. Considering how light the rifle is, a good elk cartridge is going to kick, no way around it. The .375 JDJ in potent medicine for heavy game with the right bullet and would be a better choice than the .375 win if the range might be out there a little. I would say that the recoil of the .375 JDJ in my contender rifle is on par with a 300 win mag in a heavier rifle. Stout, but by no means bad, and worth it, considering what a light weight handy rifle the contender is in that configuration. Would I enjoy shooting it on the bench for long periods of time, no way.
If you have an encore, then your choices are limit less and anything from a 7-08 on up should suffice, depending on the size of the Elk, range shot and bullet selection.
Elk are tough as hell and I agree with the shoot till they drop theory. I shot one with a 30-06 3 times all in the chest just to make sure. As said, they live in nasty country and chances are that if they run off, many hours will be added to the recovery time. Id rather have enough rifle, but not so much rifle that I am scared to shoot in accurately. That is a choice you will have to make. I think if you can handle hot 30-06 loads in an average weight gun, the .375 JDJ will not be hard for you to handle. If you have an Encore, then I would probably choose the trusty old 30-06,its hard to beat.
Good hunting,
Brian