I've been playing professionally for many years and have always had to look at my hands for stride piano, though that wasn't my specialty. Nevertheless, I have a hard time believing that all but the very best stride pianists must look at their left hands pretty regularly.
That being said, when an accomplished jazz pianist I was taking lessons from several years ago challenged me to play my scales without looking, I was shocked that I kept missing my thumb tuck, no matter the scale. Turns out I had never gained a feeling in my hands for the tuck of a close note (i.e. F# -> G in G maj in RH) as opposed to that of a note away (Bb -> C in F maj RH). I was subconsciously relying on my sight to play it correctly. A few weeks of dedicated practicing resolved the problem and changed my habit, but it was unnerving nevertheless.
In your case, and for players in general, my rule-of-thumb advice is that you should be able to play without looking as long as your hands aren't changing positions drastically. I think it's always normal to look down quickly if you're moving a 5th or so without anything between.
If you are butchering your pieces, as you say, when attempting to play without looking at your hands then you need to slow it down in to the zone where you are playing correctly. (You'll probably clean up a few other things in the side!) Although this is excruciating at first, make yourself stay slow and only speed up only once you're satisfied with how you're doing. It will go by quickly in the end, and your skills and performance will improve.