Can you post a pic of your "scope"? I'm curious how you managed that; it shows a level of resourcefulness that is not very common.
Free or cheap generally means older, which means it will have significant mass which can make shipping an issue. DSOs have been a paradigm shift, but that doesn't mean you can't get useful info from an older analog scope, certainly at the initial levels.
Fixing old test equipment can be an inexpensive way to outfit a lab and a great learning experience. Nothing teaches better than working through an instrument to understand WHY it is not working like it should and identifying the root cause.
Do you need to be able to do this to be an EE? no, just like you can be an ME and never touch a wrench, but it will make you a better one.
Building equipment is another avenue, particularly if you build your own design. Packaging can be a problem, unless you have access to a maker space or machine shop. Power supplies are always a good candidate particularly if you're good at scrounging. Here though, it's probably less expensive to buy a dead HP triple output power supply off of ebay and fix it.
Still, you can build lots of useful gizmos based on small processor boards and they can be very cheap.
If you're associated with a college or univerisity, sometimes you can get samples from the major manufacturers.
Anyway, I gotta move my lab in a couple months and I have to weed out some stuff, drop me a PM and we'll see if there's anything you can use.