What seems to help is intently listening to other players with a good swing and imitating what they are doing. Although there is a broad spectrum of swing feels you can check out some of the great ones. Some of my favorites are Cannonball Adderly, Sonny Rollins, Count Basie, Chet Baker, Wes Montgomery just to name a few.
Pay attention to the way they can play with a solid eight note feel. This maybe better to familiarize yourself with the slower tempo tunes. The notes can vary from a legato even eight notes ex. tu-tu, tu-tu, tu-tu, tu-tu or you can pull and push the eighth notes with a shoo-be, doo-be, shoo-be, doo-be.
After all you can appreciate jazz and swing is a language .... that's why they called it be-bop in the first place. It's an aural language.
Translate the eighth note language into oral syllables like ooh-bop sha-bams and sing em that way when you are playing your horn and the swing should take care of itself.
Learn the aural language from the guys that can already swing and imitate their feel. Imitation is the best way to learn the language and then make up your own stuff.
I found an interesting statement made by Dizzy Gillespie where he said he thought of the rhythms first and the tones second. So in jazz it seems as though rhythm is king. So if you think of the rhytmn first and play with a good swing feel you are on your way.
hope that helps.
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