Why doesn't my power cable light green and properly charge battery?

About a year or more ago I noticed that the little light on my magnetic charging cable no longer lit up when attached to the computer.

When I jiggle it, it will eventually light up and show either green or orange (depending on how much of a charge it needs). However, it will very shortly go dim (and sometimes go totally out) and sometimes it doesn't charge my battery.

This has caused the lifespan of two different brand new Apple laptop batteries to be very short. I've cleaned the connector area of all particles and looked all over on the web for what to do about this, to no avail.

Can I replace the "charging port" on my laptop? If so, how do I do that and where can I buy that part? I have already tried replacing both the battery itself and also the charger. Both were official Apple parts purchased from the Apple Store and neither option has helped with this charging problem.

Please help.

Sincerely,

Edwin