iTunes (Music.app) not converting song entirely / skip to the next one when playing
As a troubleshooting step I would start with booting into Safe Mode to see if the behaviour is replicated there, and also to see if the behaviour continues again afterwards when you've logged back in normally.

Boot into Safe Mode

Follow these steps to boot your Mac into Safe Mode:


Fully shut down your Mac
Restart your Mac
Immediately press the Shift key and keep it down 
Let go of the Shift key when you see the login window (NOTE: If you have FileVault enabled you may need to log in twice).
Take a note of what happens (i.e. can iTunes play and convert full songs or not?)
Now exit Safe Mode by restarting your Mac as normal
Log in normally and test again


Once you've booted into Safe Mode, let me know how you went and we'll go from there.

Test with a new user account

Okay, let's now test to see if this behaviour persists in a newly created user account.


Go to Apple > System Preferences...
Select the Users &amp; Groups preference pane
Click on the plus sign at bottom left (you may need to unlock the padlock first)
Now add the details for the new account and click on the Create User button
Once created, click on Login Options 
Ensure that automatic login is off
Also ensure that the Show fast user switching menu is enabled
Exit preferences


Now log into the newly created account (there should be an option at top right of the menu bar), download an .mp3 file and test iTunes again.

Let me know how you go.