Tube amp $500 for LCD-2

I'd recommend the Little Dot VI+. A great tube amp for headphones (especially if you get balanced headphone cables). I use one as both a preamp and headphone amp.

Eastern Electric Minimax integrated.

Sorry if I made a mistake, but if we are talking about a headphone amp and not a regular poweramp for speakers, my recommendation, of course, is not correct. Either way, $500 is a very low budget for what you are asking and any modern design will most likely, not aim for a very dark, classic sound, but something more flat. If you can't find anything you like, the next best thing may be an EQ. Not a perfect solution, but it may help.

I know a few SS amps that have the tubey sound characteristics and works well with the Audez'e. Unfortunately they are a bit out of your price range.

this is best asked on head-fi.org if you haven't already; it specifically caters to headphone audiophiles

I forget what LCD stands for but given your price range and the type of sound you are looking for, I would go with a vintage amp. Something like a heathkit or dynaco should do a great job for your needs.