When you said starter "sound like it's not engaging". Keep in mind that the pinion gear is manually engaged by your foot at the same time starter button is depressed by the mechanism. So go under hood and operate the lever by "hand" (a wooden stick, whatever) and see if starter gear is sliding into flywheel gear. Visually. Not by "sound". When the pinion engages the flywheel, starter spins..... the flywheel. I don't know which part of that operation is or is not happening. Visual is the only way.

You have several issues as you know. First you need to be sure motor spins reliably so as to tackle the no-start. Chances are your battery is no good. I would leave the starter in there until I was sure it was not a voltage supply or gear lever or starter button switch issue. Do you have a button switch mounted on top of starter? If so focus on that and the battery.

If you still must remove starter:
I assume the pinion gear is retracted. I assume all starter bolts removed. If so, just bang on it side to side, up and down with wood or rubber mallet. As said, it's rust locked or dirt locked probably. If it's all old 1937 stuff, everything will be a challenge and take special care. Save all parts to be rebuilt or matched at parts counter.
3B has given you direction on cables. That is important. The "success" you had could be the cable cleaning or could be a charged up battery floating to bad again. It could be the other items mentioned including a worn out starter. It does not sound like a bad starter "motor", the way you describe. Not sure. Study the symptoms a little more and report.

Sometimes it's a good idea to remove spark plugs while dinking around with starter. Be sure to place in neutral and block the wheels for safety. You don't want to screw around with it trying to start while you are getting it to crank nice. Starting is later. Keep key off. Put the ether away. If motor won't spin, fuel pump won't pump, carb won't work, spark won't generate, etc.

Right is right, even if everyone is against it, and wrong is wrong, even if everyone is for it. - William Penn