1. you can try to audition brands that are famous for making turntable, these brands tune their cdp based on vinyl sound, u could look around linn, roksan, rega etc. u could get them used under a grand.
2. think about what it is about vinyl sound you like, you want natural, cdp can do natural, you want laid back, warm, dynamic, etc i think cdp thesedays can do pretty good job on those things, but you can't expect microwave food to taste the same as can food, both are processed but done differently.
My take on this is, the sound is dependable on the type of disc, some certainly sound better with vinyl(recorded back in the early days before digital was introduced), now with sacd/cd format, when matched carefully, u can have a top notch system with cdp, like others suggested, it might be harder to get the sound u want upgrading/changing cdp with that budget, than the rest of your system.