Do I need the Developer Program to sign an app for my own use?
You do not need to join the $99 Developer program.
There are two cases:
Intel Mac: You do not need to sign the app.  And there may be an advantage to that as it then doesn't need to be checked for certificate revocation or notarisation.
M1 Mac: It must be signed, but you can do that with an ad-hoc signature with codesign -s - appname.app
Have a read of this from Howard Oakley.  It is focused on avoiding OCSP (certificate revocation checks), but has the detail you need which is not very much!
There are Apple documents too.  For example Open app from un identified developer.  Basically saying to Control-click and open.