Want to get into Analog

Actually, the only reality that counts belongs to the OP ;)

I’m thinking of taking a plunge into analog sources by picking up a vintage DD turntable.

You must be an experienced user to understand what you’re buying when it comes to vintage DD. Or you need someone you can assist you.

There seems to be an endless supply of vintage tables available on eBay and CL.

Post the links, what do you like ?

Which models would be good values for under $1000 (total budget for turntable, arm, cartridge.

Your total budget is too low, but if you can exclude a cartridge from this budget i think you can buy some nice vintage DD turntables like Denon DP-80, you will have to make your own plinth for it if you can’t find this drive in original plinth, then you can add Denon DA-401 tonearm. This combination could be under $1k, but then you need a high compliance cartridge, i think for $300-400 you can find a good one (i could help, got some nice spares).

You can use the same scenario with Victor DD turntables, find a TT-81 drive, make custom plinth, add Victor UA-7045 tonearm. And fidn yourself a mid compliance cartridge like Victor X-1IIe for $300-400

Anyway you need at least $1200-1500 for everything.

And if you don’t know for $1400 you can buy brand new Technics SL1500c with build-in phono stage and pre-mounted cartridge as a package deal. This is a brand new turntable, released in 2019. Maybe it’s better ? Technics motor is superb, turntable made in Japan. All for $1400 including phono stage and cartridge. Think about it. 

Technics SL-1600MK2 if you can find one.  It is fully automatic, and unlike the vintage SL-1200, it was not designed to be a DJ turntable.  I’ve had one for over 30 years and its still going strong in my secondary system.

The Technics SL-1200mk2 is a risky proposition. Most of them have been thrashed by rap-“DJ”s and will need all moving parts replaced. As OldSchool said, a fully automatic SL-1600mk2 is an vastly superior turntable, but very rare as are its SL-1700mk2 semi-automatic and SL-1800mk2 brothers. You could start off with an inexpensive AT95E to stay within your budget, and over time upgrade to an AT VM540ML, better tonearm wiring, interconnects, phono stage and KAB fluid damper. 

Why does it have to be vintage? For your first TT and for simplicity, you can get a new Rega P3 with an Elys cartridge within your budget. For a little more you can upgrade the cartridge to an Exact 2. It’s a great starter table.
Also, I would be very reluctant to buy a "vintage" turntable on Ebay.
Audiogon or US Audio Mart, probably yes.