Soap is a good idea, trouble is it is not much help when out and about, unless you also carry the kitchen sink.
I have some 99.7% alcohol at home (IPA for cleaning pcbs) but I believe 70% is the recommended alcohol content for sanitizers. Stronger can cause skin damage, if used too often.
I’m not surprised by the shortages. People are being told to wash more often and to use sanitizer when washing is not possible. Arguably, the shortages just show that people are taking notice, using more soap and more sanitizer, which is a good thing.
One thing that sometimes annoys me is, after washing my hands in a public facility, I often have to handle a door knob to get out again, after which I want to wash my hands again...