If I understand correctly that the problems began iafter february, when you had the cassette and chain replaced, I wonder if something was done wrong or poorly, or even just different. For instance:
- Does the chain width (referred to by the number of rear sprockets it is intended to work with) match the rest of drivetrain? Including the derailleurs (width of shift cage), chainrings (width of rings and spacing between them) and cassette (number of cogs plus same width/spacing concerns as with rings). One of the benefits of friction shifting is that you can usually get away with mixing and matching a bit. For instance, a 9 speed chain on rings spaced for an 8 speed era drivetrain, or a 10 speed cassette. Or a narrower 10 speed chain working on an otherwise all-9 speed system. Etc., etc. But if you're not super comfortable with friction, you're going to be better off if everything matches.
- is there a wobble in the cassette when the wheel is spinning? Maybe it got installed with some dirt behind it, or with the lockring not fully secured?
Otherwise, I agree with many of the above suggestions as place to start and things to rule out, and that it doesn't strike me as a "shifter" issue at all: Hanger alignment; possibility that you're simply worn out the chainrings or rear derailleur. (The former woul appear as "shart-fin" tooth profiles, and the latter would be evidenced by any play in the mechanism if you try to wiggle it.)
Lastly, make sure your drive-side crank arm isn't working loose from the bottom bracket spindle.