You are certainly not an idiot. It does sound like asthma. I was diagnosed in my 50s after a similar cold/cough that lasted six months or more, though I had a long, long history of chest infections and nobody really joined the dots up because I rarely wheezed or had difficulty breathing. I'm 61 too.
Expected peak flows vary according to age, sex, height and weight, so I can only be approximate but your initial peak flow of 250 was quite bad for a woman of your age, and the levels you are reporting now aren't that great.
Don't be afraid of Ventolin/salbutamol. You can't overdose on it. Too much just makes you a little light headed. There's quite a lot in each canister, but the only real risk is that you run out. Personally, I have built up a little stockpile to make sure I don't. BTW, the NHS pays something like £1 for each Ventolin inhaler so it costs peanuts anyway. (Some of the other inhalers are pretty pricey but Ventolin is cheap as chips). It's not a long-term treatment, and as others have said you may in time be able to dispense with it, but that's in the future. As I said to DW this morning when she asked if I had some with me, if I am dressed there's a can of Ventolin in my pocket and if I'm undressed there's one no more than 10 feet away,
But if you are struggling to breathe for more than a few moments you need urgent medical attention. Asthma kills. The fact that you haven't yet had the formal diagnosis because there is a waiting list for spirometry doesn't change that.
If you are coughing non-stop you need medical attention now, from a doctor. Out-of-hours should be able to deal with it but if they can't go to A&E.