Technique to bake donair meat in an oven
Disclaimer: I have not made Donairs at home.  However, I think the same principles apply to döner kebabs, gyros, shawarma, and really almost any fast-food meat.

I rather suspect that the problem is that you're using an oven.  Donairs, döner kebab, gyros, are all similar and are all intended to be cooked on a vertical spit that puts out upwards of 50,000 BTU.  Although that cumulative heat is usually spread across a few individual broilers and designed to cook a very large amount of the stuff at a time, the fact remains that this type of meat does best with quick, high, direct heat.

The grill on a gryo machine is much closer to a BBQ or stovetop than it is to an oven.  Ovens impart heat very gradually by comparison, giving the meat plenty of time to dry out.  And the oven does dry meat out; so much so that it's actually possible and relatively straightforward to make jerky in an oven, but that's a topic for another day.

Meat starts to release all its juices at 130° F / 54.5° C.  If you hold it in an oven at 375° (F or C, hardly matters at that point) for an extended period of time then it is, quite literally, going to turn into jerky.  By the time it gets above 150° F it is way overdone.  Note that at 180° F or 82° C, the collagen breaks down and converts to gelatin, which gives that really succulent and tender texture found in slow cooking, but in order to accomplish that you need to find a way to preserve moisture while the temperature rises (i.e. by brining, braising, or steaming).  More on this at the Science of Slow Cooking, if you're interested.

But, I suspect that you're not shooting for slow-cooking, so I'm going to suggest the next-best thing, which is to grill the meat instead of baking/roasting it.  Just don't use the oven.  When you grill it you are just searing the outside; the inside will not have a chance to get overdone or eject too much moisture (although you will still lose some, but gyros/donair meat is supposed to be a little dry).  The cuts tend to be so thin that it should only require a couple of minutes on the grill.

Grilling the meat should get you very close to the kind of consistency you get with a high-heat spit.  Most "street meat" I've eaten (not that I eat a lot of it!) has the consistency of grilled meat; well-done on the outside, medium or medium-rare on the inside.

If you insist on using the oven then use the broiler, which is also very similar to the vertical spit (although it won't broil the meat as evenly).  And if you insist on using the oven but not the broiler then I can only recommend that you use the highest temperature possible and take it out a few minutes before it reaches the desired tenderness, because even thin slices of meat will continue to cook and dry out for a few minutes afterward.