Thread not catching/skipping problem?

I've recently been attempting to learn to sew on an old singer machine (1964, 337 model) given to my mum years ago, it's seemingly in spotless condition but it has a few problems.

It likes to skip stitches and only stitch the left movement of the zig-zag stitch on lighter fabrics, it works perfectly on heavier cotton (think combat trousers) and also on plastic bags fused into 'material' but just will not work on material like thin t-shirt stuff.

Uneven/fluctuating tension and stitch length is also a problem overall, even when tension has been set correctly it can seem to change during sewing for no reason, sometimes perfect, sometimes pulling through one side or the other slightly.

I've replaced the needle (which helped very slightly), I'm very used to threading it but have checked multiple times with the manual to make sure, have I checked the needle is in correctly and even tested the if the sewing foot pressure was affecting it. I've also tested different tensions and stitch lengths, some working for a very short time then failing again.

Also very occasionally when I press on the power the motor will run without moving anything, not sure if that would be relevant.

I'm now at a loss, do I just need to improve my sewing skills, does it simply need some oil (and if so, what type is best? do you really need to use the Singer oil?) or are there weird goings on with the timing or other mechanisms?

Thanks in advance for any advice.