In all honesty, many natives would use either without thinking much about it.
However, if we do want to try to find a difference, I would say this:
I'm feeling sleepy 💤 - makes you mentally locate the action slightly further back in time
( something you are perhaps aware of already)
I feel sleepy 💤 - makes the idea more immediate for the speaker amd listener.
Definitely though, 'I feel sleepy' is better and more common to use!