Not true. You can run along the perimeters of most municipal golf courses without permission.
Just be courteous and stay out of golfers way. They're public city parks.
As a runner who's run on many golf courses and a golfer I don't mind runners running around the course when I'm playing as long as they stay on the perimeter.
If you get hit with a golf ball that's your problem, not theirs. You run on the course at your own risk.
In Albuquerque we have many elite runners who do so.
Now, if you paid a greens fee I suppose you could go ON the course and run up and down the fairways if you wanted to. Maybe carry an iron or a putter with you to look legit!
Joking aside, stick with late afternoons on weekdays when there are fewer golfers on the course and you'd be fine.
You may get a volunteer marshall to ask you to leave but they have no legal authority.
That said I wouldn't try running on any private course for obvious reasons. That's trespassing.
And definitely don't attempt to get onto Augusta while you're in Georgia!