New Cox Ultimate 150Mbps, but I'm still at 100Mbps

Can you post the model numbers of the different equipment? Specially the modem and router.

The first thing you want to do is check all the connections between the devices. All ethernet have to be CAT5e and above and the duplex has to be set to 10/100/1000 (unless auto-negotiation detects it). Many modems and routers have color corresponding lights to tell you what connection rate is use. This is especially important if the cap is exactly 100Mb.

I think your biggest problem is going to be your 4 channel modem.  Those are effectively obsolete for Ultimate Tier service and less than ideal for Premier.  I think it may be time to upgrade again.

The modem is the Motorolla SB6120 and the Router is the Netgear R7000, all using CAT6 cables. The modem and router lights are appropriate, blue for the modem on the up/down streams and solid on the router. The NIC is set to 10/100/1000.

What about the link ight on the SB6120, is that blue or amber?

If that still isn't working then it could be congestion in the area. The SB6120 maxes out at 171Mbps hypothetical, so in real life, I doubt you will ever see 150Mb. Modem might not be your only problem, but I think it is A problem. The SB6141 is the sessor to the SB6120, so might want to check that out but any modem marked for Ultimate will work. See here.

It is blue as stated in the previous post.

The technician who came by when I first upgraded to Ultimate said that my modem should be able to handle it. If it is congestion do I simply just wait and see?

Once your provisioned rate exceeds 50% of the modem's capacity your speeds will start to become more significant.  If you're on Ultimate that puts your provisioned rate at almost the maximum of the modem which is not good at all.  This is the reason why we made 8 channel modems mandatory for 150Mbps service when signing up.  Because your market launched Ultimate at a lower rate initially you got grandfathered in and evaded this basic requirement.