I think there are actually two separate issues here. One has to do with the bike, and the other has to do with expectations. You like your Oxford and the style of riding it lends itself to. Your previous commute was perfect for it, - short and relatively flat. You could get to work without much effort and no need to change clothes.

Though your new commute still isn't a terribly long one, it's a completely different kettle of fish. Regardless of what bike you use you're going to have to treat it accordingly. Getting you and any bike up hill is going to require some exertion. If the hills are long and steep enough, sweat is going to be part of the deal. You can decide how you manage that. For example, choosing some different clothing can prevent more of it from getting trapped next to your skin.

I do think lower gearing will help too, but in the end you still have to get to the top of the hills. There is no free lunch.

A few summers ago, I was climbing one of the few hills on my commute. It was hot and humid. I passed a woman wearing a light summery dress riding a cruiser/city bike. She could have been pulled right from the pages of "Cycle Chic" except that she was covered in sweat. It's not a look that I mind but I doubt it's what she intended. And of course I was sweaty too but a shower and change of clothes were waiting for me.