Take the drive apart. Clean the heads and rails with alcohol soaked Q-tips, then apply some lithium grease to the rails (make sure not to get any on the heads). Even just cleaning gunk off the rails with alcohol will allow them to slide more freely, and may improve the error rate. After 35-40 years, all these 5.25" drives need to be cleaned out and lubed. That's just to eliminate the obvious.
Can we assume that you're using 360K disks and not 1.2MB, and that you've verified the disk is good on the other machine you're copying software from? I've been doing the same with a PC-Jr I recently acquired. I thought at first it was the drive that was the issue, only to discover that the drive itself is good, and half of all my disks are bad.
Also, make sure the drive that came with your EX is actually a 360K drive, and not 1.2MB. You can use a 1.44MB drive in lieu of a 720K (one of the drives on my HX is actually a 1.44MB, even though the machine can only recognize it as 720K), but a 1.2MB drive can NOT be used on that controller.
I'm jealous that you have an external Tandy floppy though. Those are hard to come by these days.
---Kyle