Ask a dozen modellers about their A/B cleaning routine and what they use and you'll probably get a dozen different answers. Here's mine.
I've tried most of the branded A/B cleaners over the years and none of them work to my satisfaction (even when used with their own brand paint). For any given cleaner, you will find some modellers who swear by it, and yet others who will swear at it! I only use acrylics and now I stick to IPA for my Vallejo paints, and Mr Levelling Thinner for all my glass bottled stuff - mostly Tamiya. You will develop your own routine and it's worth your while to try a variety of cleaners until you find one that you're happy with (Clearly the alcohol you're using doesn't work for you).
I suspect it's very expensive to use MLT as a cleaner rather than a thinner, and I daresay some will agree with that, but all I care about is that it works for me. My cleaning routine is now down to less than 2 minutes with no need to dismantle any of the components of my A/B, although I do strip it down and run it through an ultrasonic cleaner (plain old tap water with a couple of squirts of cheapo liquid hand soap) every week or so depending on how much I've used it.
Hold it under the cold tap (or use a sprayer and bucket like Steven000 does if you're not next to a sink) and rinse out any remaining paint. Then a few squirts of water through the nozzle while it's still under the tap. Empty out any water and follow up with half a cup of cleaner and spray half of that through. Use a decent paper towel ('cos the cheap ones can leave tiny fibres which can block your nozzle) and wipe around the inside of the paint cup using the cleaner left over. Add another half cup of thinner and back flush until it stays clean. You may need to repeat with more cleaner. Once it stays clean, spray another half cup of cleaner onto a paper towel and check if there's any colour still coming through. Tricky if you use white paint or clear varnish on a white towel- but what the heck, the world ain't perfect!
My go to spray paints are Tamiya but if I know I'll be using Vallejo next, I'll run the A/B under the cold tap for a few seconds to get rid of any cleaner that might be left. I've had problems in the past with paint going thick or even fizzy(!) which I've had to put down to contamination from cleaners left in the A/B.
Whatever you do, make sure you clean your A/B after every painting session. DON'T do what I did a couple of days ago and forget to clean it. I found it full of half-dried paint the next morning! My normal 2 minute clean took me nearly 40 minutes!
