There are lowering seats for the DRZ that will chop an inch or to off. As well, dealerships are, I believe, able to lower the suspension an inch or two (this is done by using a shorter stiffer shock, so it won’t be as plush a ride). I’d call a local suzuki dealership to find out what options they have.
If you need more speed than a 250 you might want to look at the Versys. It has a fairly long travel suspension, with a seat that is only slightly higher than sport bike. The tires are primarily road, but handle gravel and access roads.
Similarly the BMW F650 is short by offroading standards, but will handle light offorad duties.
Neither of these options are terribly cheap, as the Versys is new, and the BMW is,well,a BMW.
I’m not sure just how “off road” you are, but if a car can get down the road without special considerations, most bikes should be able to as well. Although, I wouldn’t recommend medium or higher speeds on such roads. I’ve ridden an old UJM down pitted gravel roads and it wasn’t fazed one bit at 35mph (I however was scared out of my skull).