That annual is eight issues away issues away in my "Captain America" discussion. (It's written by J.M. DeMatties, though, not Jim Shooter.) It features the three Caps listed by Pete above (i.e., two between '46-''49).
...Maybe both DeeMattesis and Shooter did stories involving the Truman-era Caps. One of them was killed off in-story by Roy Thomas - The story I remember involved on of the Scottish? Caps putting on the Cap outfit again to have a last, final, battle with the cancer he had - The Big C ".
...U think my phone's autocorrect function wrote " Scottish? Caps " above! What I meant, I dunno...
The one killed off in-story by Roy Thomas Was William Naslund (formerly the Spirit of '76) in What If...? #4.
The one dying of cancer was Jeff Mace (formerly the Patriot) and the story you describe was Captain America Annual #6. Mace succumbed to his cancer in Captain America #284, which is nine issues away in my "Captain America" discussion as I post this.
Tomorrow morning (12/1/20), Turner Classic Movies is showing Zero Hour (1957). This movie with its reluctant pilot and food poisoning was an inspiration for the movie Airplane. I've always wanted to see it and it's not on DVD.