The Austin print catalog lists 8 thornless ("or nearly so") Austin roses:

A Shropshire Lad
Heritage
James Galway
Kew Gardens
Malvern Hills
Mortimer Sackler
Queen of Sweden
Snow Goose

When I cross reference this list with those roses he lists as hardy to zone 4, only one remains: A Shropshire Lad.

I haven't grown this one, but remember reading that it is pretty disease resistant and did well in part shade.

Here is a link that might be useful: A Shropshire Lad

Brother Cadfael is my least thorny Austin and he grows very large. Three of the Bro would probably swallow my small yard. In your zone, I'm sure he would be smaller. I'm not sure if he's hardy to zone 4, though.
Diane

I've grown a few Austins, I'm not sure how they will grow in your zone, but here's how they performed for me. Also, i prune minimally, so mine may be a bit larger than most. They are all nearly thornless. I also do no winter protection.

Heritage - good bloomer, good disease resistance. I've grown this one from about 20yrs and only had 1 off year. Mine peaks at about 7' high x 6' wide. Not much winter die-back.

James Galway - not a great bloomer, really great in the spring, then sporadic after that. Good disease resistance. It took mine a couple of years for it to bloom enough for me. Mine's really big about 15' high, 8' wide. i like it.
Not much winter die-back.

Brother Cadfael, not a great bloomer for me, not good disease resistance, weak stemmed, it's gone.

I never saw any thorns on Mortimer Sackler. It's highly disease resistant and a nice plant in every way I know of. The picture on the David Austin website accurately reflects the delicate looks of its soft pink blooms.

Ours got rose rosette disease so we didn't have it for long--it had just gotten started on its second season--but from what I saw, it's not terribly bushy and would prefer to be used as a short climber. The canes are quite flexible too, and that's the way we were planning to use it. You probably could peg it to the ground and use it as a lower ground-hugging rose if you wanted something lower than, say, the 8 feet or so reach of its long arms. David Austin's site says you can use it as a free-standing bush too, but we didn't have ours long enough to know for sure how it would look that way. But it was attractive, just the way it was as a youngster, as long as you weren't expecting a very bushy plant.

The rose does want to bloom, and will do so shortly after you plant it in the spring (from a bareroot), but it's not as shade tolerant as we needed in that too-shady spot in our yard, so once the deciduous trees leafed out, the rate of blooming and growth slowed down in a hurry. At the time ours got rose rosette disease I took some pictures and also discovered that ours had rose mosaic virus too, so the rose had some distinct disadvantages here and languished. Still, in its time here, it didn't blackspot, or at least not so that I noticed. In a sunny location with a virus-free version, I'd think the rose would be close to bullet-proof. David Austin is replacing ours soon so I think we'll have a chance to check that out; I'm looking forward to it!

It's a rose that I've been surprised not to have read more about, given its high disease resistance and thornlessness. I haven't read of anyone trying it who hasn't thought it was a good rose.

Best wishes,
Mary

Brother Cadfael is the only Austin I have that has very few thorns. It took a while to produce lots of flowers but does well now. Gets really tall here, so I just keep pruning to keep it in bounds for the space it's in.

I was at the huntington today for a camellia show so I passed by the English rose beds looking at thorns since they didnt have any leaves yet. The hands down winner of smoothest canes was Heritage.

Thank you for your replies, and esp. to Kittymoonbeam for taking a look for me at Huntington. When I used to live in NorCal, I flew down for a day trip to Huntington just to see the Tamora mass planting. It was gorgeous!

I had already planned to put the following non-thorny Austins in this year:
A Shropshire Lad
Heritage
James Galway
Brother Cadfael

In addition, I've also selected the following on reports that they weren't "too" thorny
Spirit of Freedom
Jude the Obscure
Abraham Darby

Are there others that are not "too" thorny that I should consider?

I'm also surprised that no one mentioned Evelyn--that's usually reputed to be not thorny, isn't it? Though when I grew it many years ago, it was not vigorous at all and got to all of 2' tall before petering out.

Two early Austins, Wife of Bath and Charles Darwin, are not very thorny so far, although they're still quite young plants. The equally young Queen Nefertiti, however, is very thorny, so age may not be a factor in determining thorniness.

Ingrid

I have five Evelyns and they are pretty thorny. They are all fairly large, too, except one. My huge Jude is quite thorny and has managed to inflict some damage when I've had to work around him.
Diane

Evelyn and Jude the Obscure are prickly roses, but their thorns are well spaced out. Therefore, you can hold on to a stem of flower without having your glooves on.

The real killers are: Gertrude Jekyll, St Swithun or Sharifa Asma and Felicite Parmentier (old garden rose).
Those roses require glooves on when you hold a stem.

Windermere has few thorns, good health, a lovely scent and a nice creamy color (not stark white), but the blooms don't last particularly long. I love it, though.

Tamora, on the other hand, is the vegetable equivalent of a shark.

We did the pruned rose tour of Descanso between bouts of rain and saw a pretty smooth Queen of Sweden. I got my DA order and the Snow Goose rambler had no thorns as well.