my RCA cable will not allow a recording via the headphone input on this computer
Actually, if you have an old enough Mac, it has an actual Stereo Line-In. Circle with two black arrows.
If it’s a smaller Mac (13" ??) you have one hole and you have to switch it between Stereo Line-In and Stereo Line-Out. One service at a time. All perfectly valid.
Does the new machine have USB ports?
If it does, then the Behringer UCA-202 is for you. It’s a bi-directional stereo adapter with headphone monitoring. That’s my sound mixer on the right.
There is a new version of Audacity on the way shortly that will be Catalina Compliant.
On the older machine, run Activity Monitor. Desktop > Go > Utilities > Activity Monitor. I keep a copy of that ready in my Dock. See if there isn’t one thing, application, or service hogging the machine. Do you use Adobe products? Their various license servers can be a chore to deal with.
But I do expect the older machine to work. I used to regularly get one of the department managers out of trouble by actually closing all the apps he had “napping” in the dock. Application with a little light under it? Right-Click or Control-Tap > Quit.
Browsers can leave trash lying around. Safari > History > Clear History. Be careful with this one because your auto-logins may vanish and you’ll need to know all your passwords and login info. You should have that written down anyway.
Do you have a jammed desktop? I know how tempting that is, but you shouldn’t be storing many more than about 5-7 things on your desktop at a time. The machine has to take the time to actively manage each and every “thing” on your desktop. I have a folder called “Desktops.” Every so often, I gather everything up, dump it in a folder, date it, and move it to Desktops. Use ISO dates. If 20200502 bothers you, put the dashes in. 2020-05-02.
Use Spotlight (upper right search tool) to find things. Pause on a Spotlight entry and it will open a preview panel. If you press and hold Command while Spotlight is active, it will cycle through name and location.