New cable modem, wifi router, telephone service ?

I would go with an OBiHai or OOma device and Google Voice for phone.  One time purchase and, at this time, no monthly bill.  That would enable you to use just a stand-a-lone modem and your Linksys router.  As you can tell, I'm not a fan of gateways.  Caution, depending on your Cox contract and bundle, dropping phone will have a different effect ... I would check with Cox before cutting the phone cord.

"I was having trouble with our Cox supplied Cisco DPQ3925 "

"Cox replaced the cable modem router with anew one"

Could you explain this? Is the new one the DPQ3925 or something else?

"buy separate wifi router which is now installed and working. (Linksys WRT1900 ACS)"

You buy a separate router when you have a separate modem. But you said Cox replaced the device with a new "cable modem router" which isn't a separate modem, but a modem router combo. You would either need to set the Linksys to AP mode or trade in the modem/router combo for a stand alone modem with phone(eMTA) like the Cisco DPQ3212. That's what I have now with Preferred but it is good up to 100Mbps.

"but finding that it randomly rejects certain devices that were previously working fine"

What devices? Like laptops and tablets? And what do you mean by "reject"?

"Was advised by Cox techs that I did not need to disable the wifi on the Cisco, and only choose the wifi on the device and sign in accordingly to either one we wanted.  Worried about conflicts there."

This isn't clear to me, but my gut is saying you got mis-information. Its all about what they call NAT, Network Address Translation. Think of it like a switch board operator at a big business. If you call/connect to someone from your phone/computer, and that person wants to call you back, they don't know your phone/computers identifier, only that of your company. NAT takes the 1 public IP Cox gives you and splits it into a bunch of private IP for your devices to use. Kind of like a company having lots of extensions, but only 1 phone number. The problem comes when you have two devices with NAT connected to each other. Someone calls for Bob, but router one only knows Sales, Billing, and Technical support, and only router 2 knows Bob is in Sales. You can rig some port forwarding and triggering to get it all to work, but it's a big pain and the extra work done but the router slows everything down. That is why its good to have one device be the modem/eMTA(no NAT) and one device be a router, like the WRT1900ACS. 

"should I/can I buy a separate cable modem for high speed internet and split off the Cox modem for only the phone service, or just keep Cox high speed internet and go back to Verizon phone?"

Depends on your signal going to your modem now. If you can cut it in half, and still have it be good, then go with the DPQ3212 for phone, split the line and use SB6183 or SB6141 and your Linksys router for data. If your signal is good, but not great enough to be cut in half, just use the DPQ3212 for your phone modem and internet modem. The DPQ3212 is a good modem, but lacks diagnostics. Also, I wouldn't suggest buying one with a battery. Many times they are refurbished and cause T3 power issues. Instead, if your worried about phone during a power outage, but a external battery. Last longer then any cheap thing they can sell you for the eMTA and makes it so you don't need to use the reset button the back of the eMTA to reset it. 

The only problem there is where do you want your phone? Do you use cordless phones with one base? Have a fax? Etc. If its in the same room as modem, or if there is a phone jack in that room, your all set. If not, will need more info on how your house's phone wiring is set up.

I am using the OBi200 so familiar with that device.  I disconnected the incoming feed at the dmark terminal block, left all internal connections.  The OBi200 does have a RJ11 and ran a phone cord from it to one of my RJ11 wall jacks ... all phones are active.

The OBi200 allows up to 4 VoIP providers.  I use Google Voice as my primary.  You can sign up for E911 service from Anveo directly through OBiTalk for $15/yr.  If you do so, it will automatically configure your device, see here.

Six months into using the device, no complaints.  I love Google Voice's ability to ring the home phone (via OBi200) and both my wife's and my cell phones.  Whichever device picks up the call first gets it.

Hi Tecknowhelp,  thanks for your clarification questions.  Answers as follows:

Cox replaced the cable modem combo unit with a new Cisco DPQ3925, same unit as before, only a new one.

The Cox tech support person said that I didn't need to reset or replace anything.  I thought I lost some of the advantages of the Linksys by setting it to AP mode?  If I replace with just the DPQ 3212, is there a way to get more than the 4 Ethernet connections available on the Linksys 1900?  I have a need for up to 6 connections.  

Pin answer to your "rejects" question, yes, laptops, iPads and iPhones that have the wifi code, were previously working, and suddenly, sit spinning trying to connect, unsuccessfully.  Wifi name and signal shown on list of available wifi, but won't connect.  Will then connect when the old wifi Cisco account is chosen.  I thought that the wifi could be disabled in the 3925, but told repeatedly by Cox tech that it could not.  

I was going that route after reading the responses, but it appears Cox in my area is currently out of the 3212 modem only units and issuing the 3925 only.  Will keep trying to get one, set up, and report progress.

TimV said:
but it appears Cox in my area is currently out of the 3212 modem only units and issuing the 3925 only.

That is unacceptable and probably untrue. Someone can send you one by mail and a technician can come back and do it right. And what, are they installing wireless gateways into peoples home who don't even have Cox internet? They don't think that might cause a problem? Why don't Cox HSI technicians get trained in basic networking? Being a technician and not knowing NAT is like being a mathematician and not knowing Pi.

And yes, putting your router in AP disables all the routing functions, so the DPC3925 can do them instead. Problem is DPC3925 is a POS and a lot of the features are disabled. That's exactly why stand alones are better, but Cox seems to be addicted to gateways. 

If they some how can't get you a DPQ3212 I say threaten just to cancel phone and get a normal data modem and then use google voice or Oonage for your phone. Better and cheaper. If Cox can't provide the right equipment for THEIR service, don't give them YOUR service.