I'd poke at a couple things. Measure how high the bottom brackets are, and how long the chainstays are on both bikes. If you can find out what kind of tubing each bike uses, that might also be informative.

Stability comes from a bunch of factors, some of which can be hard to measure. Some kinds of tubing are very stiff and difficult to bend. This can result in a harsh feeling ride (which tends to feel unstable) but the bike will also never feel like it is bending or wriggling under load (which also tends to feel unstable). Other kinds of tubing are a bit flexible, which means you get a pretty compliant ride, but can get wriggly feeling under a heavy load. The fork and seat tube angles can affect how stable a bike feels. It's customary for those angles to basically match and be around 72 degrees. Going up to 75 degrees can make a bike feel very responsive/twitchy. Going down to 70 degrees can make a bike feel sluggish/stable. If the angles don't match, some riders might feel like the bike is mysteriously not riding like a proper bike. The chainstay length and bottom bracket height are pretty closely related, and they tend to have a lot to do with how stable a rear load feels. If you have a 42cm chainstay, that will almost force a particular bottom bracket height. Go up to 44cm or 46cm, and the bottom bracket pretty much has to be lower to the ground.

I'm sure there are a lot of other factors. I think you'll end up happier with the changes you make if you get concrete about what is going on with the bikes you have .