So here are a few of my suggestions/thoughts on the subject. First I'd get one bike. I wouldn't rush into it because there are plenty of cool bikes in this price range. Prewar deluxe is where its at. Unfortunately nice, original paint, prewar deluxe bikes generally bring more than your budget and competition can be fierce. If you aren't adverse to decent restorations it opens a lot more possibilities. So here are some of my picks-in no particular order:
1939-40 CWC Four Gill. A restored bike should be around $3000-3500 and an original will be right at $4k
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1938 Hawthorne Zep--still a few nice originals out there that should be had for less than your budget
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1941 Colson Double Eagle Deluxe aka "Clipper" Originals of these are tough and expensive but I've seen quite a few decent restorations in the $3k and sometimes less range-unique styling
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1938-39 Iver Johnson Super Mobike--I've seen prices all over the place for these and a nice original will likely exceed your budget but if you can find a restored bike I believe it should come in under the wire.
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1939-40 Monark Five Bar--an nice original probably out of the question for the money but there are some nice restored out there that should be in the $3000-3500 range. For whatever reason I have seen a bunch of horrible restorations of these in some downright crazy colors.
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With any if the above at most rides you won't run into another. My last couple of pics if you just have to have a Schwinn would be:
1936-39 Motorbike--While prices for original bikes have been climbing I believe if you really shopped it you could find an original paint bike at $4k. Restored you might find decent at $3500 but make sure its right. The limiting factor on original paint will be color--hope you like black and ivory! While these were made in other colors they are highly sought after and will bring good money.
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1941 Autocycle Deluxe (canti)--again prices for original desirable colors will get crazy (like this blue one) but maroon/ivory or one of the other common color combos even with a Forebrake should be within range with coin left over.
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A couple I left off--37 Zep--not a big guy bike, 38-9 Huffman Twin Flex--if you are over 225lbs then unless you get a pair of John's springs not a good candidate. Good luck with your search. V/r Shawn