Macbook Pro headphone jack cuts in and out
The headphone jack has a connection problem - loose/broken solder joints, corroded connectors, and/or worn contacts.
This sentence confirms the diagnosis:

To fix the problems I have to wiggle the cord back and forth until the computer is satisfied.

If you have to take a physical action to solve a problem, it's a physical problem.  By wiggling it back and forth, you're making sure there's a good connection, but after a while it will lose it's connection again.
To fix this, you need to replace the headphone jack and this means electronics rework.  There are shops that specialize in this type of work and even though it's an Apple, it can be done and the shop doesn't need to be &quot;authorized.&quot;
If you decide not to fix it, you can use a Bluetooth headset or get a USB DAC that will allow you to connect your analog sound devices to a USB interface.