here is my direct experience with flex:

On any of my standard diameter tubing frames with "square" geometry like your bike ( for example, 1985 trek 620, 1984 specialized expedition, 1982 Kuwahara Custom) the frame will laterally flex a bit as you describe when loaded. Many (including me) feel there is a balance between comfort and "noodliness" of the frame, due to vertical flexiness

The bike frame will not bend, so dont worry about that... but it will "spring" from side to side, especially when climbing or pedaling hard. This can be a good/fun thing on fast road touring, but more of a hassle when slowly climbing or offroad (IMHO). Occasionally when pedaling very hard, the crank arm might even hit the frame a tiny bit, or the chain will rub on the derailleur due to BB flex... nothing to worry too much about- just sit back and feel strong

Really, the only way to circumvent this lateral flex side effect, (assuming same load and riding) is to build a frame from oversized diameter 'OS' tubing. I have two 26 inch wheeled bikes: a Roberts roughstuff and a Tout terrain Silkroad both build form OS columbus and/or reynolds steel. In addition these bikes have compact geometry, which also adds to their stiffness. Neither of these bikes will flex laterally, and they are therefore more surefooted beasts of burden, but they can be harsher to ride. I make up for that by using 40-50mm tires at lower psi, thus retaining stiffness while at the same time maintaining dampening of road vibrations.

You should be proud of that bike, it looks great. (ditch the boat-anchor surly front rack though )