Bin the chamois and get a drying towel, like sonus for example, difference is night and day.
Can use last touch whilst drying, to prevent water marks, and add shine.

Swirls - most cars are clearcoat scratched to fuck, needs to be machine polished (with Menzerna, for me), any bad ones cut back with unigrit then polished.
Need to use a claybar first though.

If swirls aren't too bad, in meantime could try a LSP with fillers, I've some Meguiars wax (blue bottle, forget name) which fills light scratches and swirls nicely.

Autoglym isn't that good, what colour is the car?