Another thing to check...
Magazines on Desert Eagles tend to work best if they "float". Unlike the other poster, my experience (with 2 different .357s and a .44) is that they will malfunction if there is pressure on the bottom of the magazine. Not every time, but sometimes. This may not be typical for all guns, but it has been for mine.
Your pistol is not short stroking. If it were, the bolt would not push the next round out of the magazine. What is happening is your gun is hanging up with the next round partially chambered, right?
How many rounds fired before this happens? Happens with all magazines? Happens consistantly? You say there is unburned powder in the chamber, a little? or a lot? Do you start out with a clean chamber?
One thing that may be happening is the soft lead of the JSP is not sliding along the chamber, possibly from the powder fouling, or possibly because the magazine is presenting the round at a bad angle.
Simplest and easiest thing to do is (after through cleaning) change ammo.
158gr is the correct weight bullet, and what the factory regulates the sights for. 125s can be made that will function in the gun, but they have to be pretty hot, and they will not shoot to point of aim. If you handload, and you have adjustable sights
fine. Otherwise don't bother.
Try another brand of ammo, and/or another style of bullet. Magnum Research recommends only certain ammo and loads. My experience is the guns will work with other brands/loads, but there is no telling until you try them out. Each gun is an individual, and my guns have all been older (what is today called the Mk I).
Good luck, and please keep us posted.