Need a new CD player. Suggestions please.

Look at the new Advance Paris X-CD9. It's a bit over your $1k budget ($1,399). Great sounding and looking.

If you want to have the greatest audio experience and also get the most use out of your CD collection you should seriously consider ripping the CDs and using Roon.
This makes them a snap to play to any Roon ready device. Also with all of the tremendous metadata resources, Roon takes you on a musical adventure every time you choose an album, song or artist. You not only rid yourself of all the distortions associated with CD transport, you also open up the flexibility of choosing the best DACs and amplifiers 
One great player that is Roon ready is the miniDSP SHD

I replied earlier suggesting the Marantz 30n but see you are trying to stay under $1000. In that range I have been super impressed with the Rotel CD-11 Tribute. So much value for the Monday and I enjoy mine very much. Not SACD. 

Take a peek under the covers of the latest new cd players at or around thousand bucks, and most look the same.  Simple, straightforward, with not much in the way of power supplies.  Take a look at sub $500.00.  They almost look empty.  And yet they work.  Keep in mind that players like the Advance PARIS X-CD7 sell in Europe for 400 euros, and are made in China.  I think that you can get pretty decent sound from many players at this price point.  I must say that some of the latest inexpensive offerings look like computer CD drives dressed up in nicer cabinets.  Just check out a Micro Center or whatever computer store in your area, and you'll see similar items for under $150.00, most under a hundred bucks, and not much better than my Dell ripper that sits on my desktop with a VPI Brick on it.  Sometimes, if you're lucky, you might find a great used and inexpensive CD player on the cheap, kind of like my Philips /Magnavox player.  Has one of the classic ladder DACS(TDA1541) and a beefier power supply than the recent thousand-dollar offerings.  

While this player doesn't have the details, depth of the newest affordable stable of machines, it is musical in ways the newer machines are not.  But that's my opinion and it won't play back SACD. But for the $100.00 I paid for it with remote all in like new condition, I can't complain.  Also, the first player I ever owned, 40 yrs ago, was based on this very same machine, so it was also sort of a  nostalgic purchase.  I'm biased.

   If you can stretch your budget, the Denon DCD-1700NE mentioned here is fantastic.  I heard it locally being compared to a Cambridge and a Marantz in the $450-650 range.  A less expensive Denon, like a grand less, was also nice.

Do they even make a (modern?) TRANSPORT that plays SACD?