Advice on Integrated Receiver

Hint 6 yes, Anthem STR, maybe Pass Labs INT 60 or Luxman 507z or 509x would be my start points. I would imagine the Pass (and know the Luxman) can let you bypass the phono preamp as mentioned above. I’m also going to throw another vote for the Hegel H390....great value and tons of current. Enjoy the search!

EDIT:  just scanned the classifieds here.  There is a Hint 6, an STR and (new entrant) a Marantz PM KI Ruby all available.  Several of the Hegel H190's too.  The Ruby is a great value IMHO.

Post a pic of your room if you can. I’d bet there’s some placement work to be done.

If you’re dissatisfied with the sound of your speakers, the two most-important factors to consider are placement in your room (and where you’re listening from) and synergy with your amplifier.
 

While you don’t say specifically why the sound of your system is not right to you, I think you’re probably right that trying an amplifier with better synergy will improve things. You also don’t say if you need/want features and versatility, or just a line-level integrated amp.  When you’re shopping, look beyond the power rating of the amplifier and look at things like higher damping factor and/or (for class A or AB amps) ability to double power from 8 ohm to 4 ohm loads. Your speakers are rated 8 ohms, dipping as low as 4.  If you are looking at class D amplifiers, look for a recent design such as one of the higher-end NAD units. 
 

Hope this helps you a bit. Have fun!

Ok dumb question, but would I be fine just using the CA CAX81 as a preamp and output the signals to a dedicated amp like the Pass Labs XA25.  Or would leveraging the CA's preamp circuity somehow be a bottleneck.   

Before I just duplicated equipment (buying another integrated amp) I figured I'd ask the question.

Preamp is a critical part of the system. It must be transparent and on the level of your amp. Unless it’s a makeshift solution until you get a good preamp, I wouldn’t use the Cambridge in that capacity if you want to go the separates route.
 

But, with a budget of $5000 you’re better off with an integrated in my opinion. So I would sell Cambridge and the 💩 and get a good integrated with or without a built in phono stage.  Any of the integrated amps mentioned above, Parasound, Luxman, Hegel and Pass will take your system to a new level. The speakers you have should let you start enjoying the music.

@bound4h what @audphile1 says above is spot on.  People suggest all the time that they just add power to their AVR and I cringe, because i've A/B tested mine and it's not really close.  

One more thing on the MA front.....when my dealer came over to final tweak my subwoofer settings, he said "I don't sell this brand but can't say anything bad about it" and further "I've seen people spend a lot more and get a lot less performance" than I was getting.  While I think part of that was a complement to a speaker that punches above its weight class, it was also a testament to a well sorted room.  So as others have asked would suggest you post a pic of that, as it's a $0 fix.