You can tell a lot about your swing by watching ball flight. One famous teacher, John Jacobs, doesn't watch anything else. That's all he needs to see to make his corrections.
You can also tell a lot about your swing by the way impact sounds. Here is where hitting off a mat is truly an advantage. When you're hitting the ball first and the ground second, you hear a click mushed up against a thunk -- like a sharp thunk -- and the ball shoots into the sky.
If you hit a little fat, you hear the thunk but no click. Too thin, and you hear the click, but no thunk.
In fact, when you hit off a mat, you should be using the sound of impact as a reality check. You can hit a little bit behind the ball on a mat and still get a decent result because the club will slide along the mat for a ways. Off turf, the ball wouldn't go anywhere.
Putting is the same way. There's a sound that only comes when you hit the ball on the sweet spot. That sound accompanies a strike that doesn't resonate into your hands and you see the ball jump off the face of the club. Paul Runyan used to practice putting by ear.
Chipping? Same thing as swinging. There's a click right on top of a thunk and the ball leaps off the clubface.
It's easiest to get this sound with your wedges, harder as you got to longer clubs, but practice with one club until you can make this sound consistently with it before moving on to the next one.
Play by ear. The ball can still go right or left, but if you're hearing the right contact, you won't see those stray shots very often. You'll get the extra distance you've been wanting, too.