Sort of know what you're talking about there Julian, as I've just bought myself a copy of the Bernny Goodman 1938 Carnegie Hall concert, which is remastered from the long-lost original '78s - and to say there's a bit of surface noise is a serious understatement! Great music though!
Since you've posted on the Classical haven, I assume you're after his more classical side rather than out-and-out Broadway.
I don't have a lot of Gershwin, but....
There appear to be 53 versions of Rhapsody in Blue (at least, from my usual supplier)! My copy of Rhapsody is on a compilation called "New World Jazz" by Michael Tilson Thomas and his semi-professional orchestra of USA's best recent music graduates, New World Symphony, on RCA 09026 687982 and is pretty good. Most of the CD though is not Gershwin, but is a collection of jazz-inspired symphonic music (e.g. Adams' Lollopalooza - which I may well have played you - Bernstein's "Prelude Fugue and Riffs", Stavinsky's Ebony Concerto etc). He uses the original Paul Whiteman Band orchestration rather than the thickened out full symphonic version that is usually done, and it sounds so much better in that version. Looks like that one's may be deleted now though (not listed on MDT, but is on Amazon and Crotchet). He's done several other versions - one playing along to Gershwin's own piano roll, which is incredibly fast - but quite exhilarating (Sony SMK42240). I've also heard similar versions from Andrew Litton and the Bournemouth Symphony Orch. (or RPO?). I also have an LP from my teen years by the LAPO with Mehta....which is one of the reasons I don't have much Gershwin - it's so overblown....lifeless...yawn.
One of the classic versions is by Bernstein and the NYPO (from 1959) - much slower and full symphonic, but supposedly magic. (now on Sony SMK63086). Also has a good version of American in Paris on the same CD.
American in Paris - the Gramophone recommendation is Chailly with the Cleveland Orchestra. I've not heard it, but looking at the performer list it has the Labeque sisters playing in Rhapsody in Blue. They are stunning (and they play fantasically too

), so that could definitely be one to check out as well. (The CD does have the rather dready "Lullaby" - all he got to complete of an intended String Quartet). I have the Labeques' piano duets CD (EMI 7243 5747292: 'I got rhythm' variations; Blue Monday; Embracable You; etc), which mixes reflective stuff with the more obviously jazzy - a lovely fusion of the world of tin-pan alley on the one hand and Debussy (and touches of Scriabin and Poulenc) on the other, although maybe a bit too down-beat for you.
For Porgy and Bess - Rattle with Willard White as Porgy is wonderful. This treats it as genuine opera rather than Broadway - although the voices are "properly" trained there's proper swing in them too when appropriate. I have the highlights version (EMI CDC 7543252)
I notice there's also a 2CD live performance by Tilson Thomas from RCA (09026689312), including American in Paris, the Piano Concerto and various bits of Porgy and Bess, with Audra McDonald and Brian Stokes Mitchell (so I would expect it to be much more Broadway than the Rattle above) - but no "Rhapsody in Blue" - maybe combine this with that "New World Jazz" CD?
Another possible candidate is Previn, although in general I would tend to put him behind Tilson Thomas and Bernstein (although he does include the Piano Concerto with the other two, e.g. EMI 5668912).
I don't really know anything about the Piano Concerto - so nothing really to recommend there.
If I were going for just one disc, I'd probably head for the Tilson Thomas Sony one, and then go for the Porgy highlights for a 2nd (although I do love that RCA disc).
[Edit to add]
After posting the above, I remembered I do actually have that Bernstein recording, although in a different coupling (with his own Westside Story and Chichester Psalms), so did a quick comparison between the two this morning. Tilson Thomas (MTT) sounds much more genuinely '20s and the orchestra is much lighter on their feet and is recorded much better. Against that Bernstein has a quite enticing "look-at-me" swagger, and the piano has a better and more believable focus than MTT's slightly soft-focus piano sound. Being the original version though the 2nd half of MTT's version does slightly outstay its welcome - it has the 44 bars that Gershwin eventually cut out (he didn't quite know when to leave off playing with that theme). I'm not sure I can really recommend one over the other - I'm happy with both.