If Conrad Johnson still made a DAC it would sound like??

Van Alstine may be close to what you seek. Definitely not as rich as a CJ product yet way less "nails on a chalkboard" sound than many of the products out there...

If you are looking for that old school detailed but musically romantic Conrad Johnson sound as your title suggests you need to up the budget and go with an Ayon lower end model. Pure Class A triode tube output designs.  Hard to find pre-owned.  I do not know how you get from CJ sound to Theta.   Otherwise, in your budget, I second going with a Bricasti M3 SE preowned. There are a number listed on AG in your budget.  Dense, palpable image, detailed, dynamic, while at the same time not bright in a negative sense in the high frequencies.  Or better yet a preowned M1 if you up the budget.  The latter more of everything you get in the M3.  

How about one of the older PS Audio Directstream DACS?

Juicy and deceptive. This made me think of Herb’s review of the Border Patrol DAC. He liked the natural sound, but JA said the measurements were terrible. Some people seem to like it. 

Switch over to the Conrad Johnson Owners Group. You may find more specific information regarding the type of DAC you are looking for that will pair well with CJ. I’ve had a Lampizator Amber 3 for a few years. There are a few users on CJOG that have a tremendous amount of institutional knowledge, not just some reviewer with 15 years of experience that you may encounter on this site.