I've got a small harem of old mics here - Shure and ElectroVoice, and an IC-7300 as well. The "444" ought to fit that category.
My own "fix" for all the old mics is to run 'em thru a preamp. In my case an ART TubeMP - which are currently going for a stupid low $45 online. Must be a close out.
I've made up a little "PTT box" with XLR input and 8-pin output. There's a DC blocking cap in there, and the box has send/receive to the preamp. Used Cliff 1/4 inch jacks to keep the audio lifted from the chassis of the box. That particular preamp has switchable 48v phantom power on the XLR inputs. I am afraid that I'd turn on the 48v by mistake and fry something, so I opted for the 1/4 inch (Hi Z) connections.
That particular preamp is set quite low, and the radios gain stays at 25 percent. Except for one noisy element I've got here.... all the old mics work great thru it.