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3/8" is pretty thin for a panel on a headboard, especially for a panel that large. It will sound too thin every time it gets hit. If your frames are cut so a 3/8" thick panel is centered, then you could make the panels 3/8 + 3/16" = 9/16" thick and cut a rabbet around the back edges of them to give a 3/8" tongue to fit into the frame. The panels will be flush with the frame on the back side, look the same as a 3/8" panel on the front side, and feel a lot more substantial. They are also likely to stay flatter after glue-up, especially if you use solid stock.
With no bandsaw I wouldn't try resawing veneer on your table saw. You'll waste more wood than you get. Best to stick with solid stock.
You also may want to consider raised panels instead of flat ones. That would allow you to make them 5/8" to 3/4" thick so they will feel even more solid and give you some additional architectural detail. I don't know what look you are after; just thought I'd throw that out there.
John
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I agree with John on every point. Your idea of cutting thin veneer (if that's on a BS) would make flat panels, you could glue the 1/8" veneer to a plywood substrate and most (including me) would suggest also putting another layer of backer veneer on the opposite side. I suppose you could get the 1/8" material while resawing on the tablesaw, but the excess waste is certainly something to consider. Would love to hear what you decide and see the final product!
I started with absolutely nothing. Now, thanks to years of hard work, careful planning, and perseverance, I find I still have most of it left.
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If you really want to go that route and resaw your thick stock why not cut the wood to 1/8" then glue the veneer to 1/2 MDF. Frame the MDF panels with solid stock, mitered at the corners, then glue the venneer over the mitered frame as well as the MDF. Raise your panel as usual. You just need to make sure that the solid stock frame is a little wider than the width of the raising. It would make a really stable panel. If you really want to go the extra mile, your frame could be short grain at the appropriate ends so that even an experienced woodworker might be fooled. If you REALY want to go an extra mile, veneer the back too.