That proves the cal output is good.
Agreed.
You do need to find the operator's manual, at some point. It out there, somewhere.
Okay let's talk about one more front panel adjustment... I think this will not be the cure, but we can talk about it... On your scope, below the Focus knob, there is a Trace Rotation Screw. I believe this to be the horizontal (Like horizontal hold on an old TV) adjustment.
If you were using a straight voltage, say from a 9V battery, you would get a straight trace, offset 9V above the baseline. If that straight line were tilted off of true horizontal, you would use this adjustment to level it. I am rather sure that it will not true the square wave. Meaning...
I think that you are on the correct track by proceeding with an internal calibration. However, this will take a good bit of knowledge. You have the manual, but it may be outside of your skills, even with the book. You will have to decide. If you did not pay much for the scope, then its a good learning opportunity.
The skew on the square wave is a common adjustment. I suggest that you watch several YouTube videos, before you start on yours. "W2eaw" does a good job on walking you through it, but you will be working on a different scope, of course.
Read the service manual, completely, first. Follow its directions step by step. Don't skip anything. If you can not true it with the adjustments provided, that will mean that there are parts issues.
You are going to need a set of proper electricians tuning tools. A regular screwdriver may influence the adjustment and you may be chancing its effects and never be able to hit the sweet spot.
I hope you do well with it. I'm glad you are saving a good piece of equipment from the trash bin. You will want to hang onto this scope, even if you buy a fancy new digital scope. Analog scopes are better at some jobs, so its always good to have one around.
Any other questions? If not, I'm going to log off.