Any idea which headphones I’d like?

Good decision to wait until you can put some on your head and ears. The comfort factor is a priority, if they aren’t comfortable it won’t matter how they sound.
With your budget I’d buy an amp, a pair of open back and a pair of closed back ‘phones. And maybe some upgraded cables.

@toddcowles understood.
Another option is The Cable Company. You can audition headphones and headphone amplifiers at home for 5% of the retail price for a period of two weeks. I believe whatever you spend on "renting" that equipment will get put into a pot and applied toward your final purchase regardless of what it was you auditioned. You can try 3 headphones and decide to apply those 5% fees towards an amplifier, for example.
Check with them.

That's a nice, healthy budget.

I too am moving away from 2 channel and into a good headphone setup.

I already have the Focal Elegias for a closed back, and I'm waiting on a backordered Schiit Lyr 3.  I plan on also adding a Bifrost 2 and a pair of Meze Empyreans around bonus time.  Hopefully that will do it for me.

Good luck.

A great source of gather tons of Idea thank you so much.

I know you said you don't want a tube setup but the Woo WA6-SE V2 is killer. Only two tubes and a rectifier.  With stock tubes in, rolled the rectifier to a Sofia Princess and the Kennerton Rognir planar Headphones, this setup seriously rivals my Magnepan 3.7. All for just over 5k. Largest soundstage for phones that I've ever heard. One must love the planar sound however, I've had maggies for over 30 years now so it wasn't odd for me to have these. I still have Audeze LCD3 which I'm going to sell, (maybe) and Grado SR125 and GS3000e.  The GS3000e will always be a keeper too.  Good luck.