Hello, thanks for your question today! I am Dr. Drew, and I am a licensed veterinarian. I'll be happy to help you in any way I can.
I am sorry to hear that Charlie is having this sort of trouble.
What you are describing, may be a condition known as "saddle thrombus", where a blood clot forms, and blocks the flow of blood to the back legs.
this causes the paws of the rear legs to become colder than the front paws,
and causes significant pain, along with inability to move or support weight with the rear legs.
This is not at all a condition that can be managed at home, unfortunately,
and even with aggressive hospitalization and intensive care,
the prognosis is quite guarded.
Typically, there is underlying heart disease which is what increases the formation of blood clots in the bloodstream,
and so even if the current clot dissolves, a new one may form in days, weeks, or months later.
many cats with this condition are humanely euthanized, sadly.