I would second billstevenson about the VPI HW tables. I used an HW19III for over 20 years with SME IV arm, which does not have a removable headshell. It is why my friend sold it as he liked swapping cartridges all the time. I don't care to but am more liable to have multiple turntables, which I have. I currently use a Linn LP12, which you may find used in your budget but it would not likely have a removable headshell either. Linns tend to be fidgity/fussy but loved by many, including myself.
You are getting a lot of Technics 1200 and 1300 suggestions. They have a lot of happy owners. I'm just not a big fan of DD. Many are.
Lots of vintage stuff out there for $1500 or less with arms that have removable headshells. Many from the 70's and 80's are great old rigs. Dual, Thorens, Micro Seiki, AR, Lenco and probably plenty of others that would perform very well for just what you want to do. It can be a fun journey...........it can also be a PITA. Best of luck.