All steel is springy, but what matters is whether you can feel that spring. Like you said, the gauge of the tubing makes the most difference.

High-end heat treated steel can be thin-walled because it's strong. That means it will bend to and fro but spring back to its original position.

"Normal" 4130 CroMo steel has to be drawn a bit thicker because it's not quite as strong. That means it may flex a bit, but you may not feel it as much.

Cheapo steel is high-tensile, and it has to be damn thick. Usually it is non-butted. Think old cheap MTB frames: heavy but indestructible because of the thick-walled tubes.

I don't know the gauge of tubing used in Surly and All-City bikes. I understand the LHT rides like a brick unloaded, but feels pretty good with a bunch of weight on it. I really hope that Surly uses thinner tubing in its roadie models like the Pacer.

Soma has some options in heat-treated Tange Prestige, for pretty reasonable prices:
Smoothie ($430), ES ($430), Stanyan ($710), and San Marcos ($930). I've seen the 1G San Marcos frames (pearly light blue) on clearance for around $750 online.

I also recommend that the OP keep his eye out for an 80s or 90s frame made from high-end heat treated steel. Since the OP is pretty light, he probably won't feel the comfy springiness on a heavier gauge frame. In the pre-carbon era, there were lots of choices in heat-treated steel and they now can be bought for relatively cheap. You can fit modern mechs on most any high end 90s steel frame.

I have two '97 bikes with thin-wall heat treated steel frames, one in Reynolds 753 and one in French Excelle Eco tubing. Each gives a better ride than any carbon or aluminum bike I've ridden.
I also have an '88 bike in Tange Prestige, but it must have thick-walled tubes because it's inflexible, even under my 230#. That bike is a rigid MTB, eventually destined to become a loaded tourer.