You might want to look at your razors. Do you use tape to hone?
If not, you may have a sharp corner on the spine and are scraping the strop with the corner of the spine.
You will feel the sharp corner, it can be smoothed with some 1-2K wet and dry.
4-5 drops of Ballistol or Neatsfoot oil on a damp strop worked in on a piece of tissue, (2 squares of toilet tissue works well) and will hydrate the leather. Do not use more that 4-5 drops at a time. Oil rubbed into damp leather will penetrate deep, oil put on dry leather will sit on top and rot the leather eventually.
With the tissue, you will feel & hear any dry spots, you can blot those spots and let the leather wick oil from the tissue pad. It may look blotchy if you leather is dry, but will even out in a couple days.
You will be tempted to add more, but don’t.
Wait a week or two and add more if needed.
I use a damp microfiber towel to wipe my strops.
Here is a good post from a leather worker/strop maker, on hydrating and oiling leather.
(Leather 101.)
Post 37 he gives his advice.
I have re-hydrated old vintage Kanayama strops, using this technique, it can take several months.