Sounds like your Dawes needs a JIS headset. The crown race should be 27.0mm, which sounds about right if the old 27.5mm one is loose, but your new 26.4mm race is way too tight. As for the cups, JIS calls for 30.0mm, vs the larger 30.2mm for ISO cups. Usually, this is a small enough difference that you can just force the ISO cups into a JIS frame, especially if they are alloy cups.

Back in the day, most of the Tange headsets were JIS. Now, most of them are ISO, but maddeningly, their JIS options use the larger 30.2mm cups. There are few if any true JIS headsets being produced currently.

If you can't source a JIS headset, it's a pretty simple operation to mill the fork crown, and ream the head tube for ISO spec parts. Probably 30-45 minute job for a shop with the proper tools.