on glass like this you have to go to town.
Dust off what you can and if they are georgian panes, either get a sponge or take the sleeve off the applicator and use that.
Nice soapy water...you will do as well with fairy as with anything else at this point...wash all of the frames and glass, if it's a smokers house you will have to be really thorough.
Bucket by the window and use your washer sleeve to wash everything down.
Use your wet scrim and mop down the frames and glass, keep rinsing out the scrim in the bucket.
Sounds longwinded, but it isn't.
Use a small squeegee and do the glass.
Dry off the frames with your REALLY WELL RUNG OUT wet scrim.
The glass and frames should at least be reasonable by now.
I'd personally recommend using a proprietry glass cleaner in a sprayer now, mist the glass and use a clean micro or scrim to just clean the glass.
If the panes are big enough then use clean water and equipment and use your applicator and squeegee (minimal water!!)
Maybe do a few windows with the first stage before you go back to them with the second stage.
The method you described would probably make even good windows appear smeary.
If you are using a microfibre or scrim to buff clean, and a strong solution in the water then you are going to leave smears.
The less you have in the water, the less there is to leave on the glass.
Ian