at the start of the cut, if the saw is wobbly, grab the spine, too, and pull it through the cut.
I get the sense that the ragged cut has a lot to do with making a stopped cut, and not the saw's potential.
I couldn't understand exactly what you were saying with your first question (in terms of what the cut would be), but it obviously makes sense now. it's an awkward cut, and I wonder if it might make more sense to mortise out some of the waste at the back of the cut by hand as you're making the cut to free things. Either way, I guess it doesn't matter too much.
Those small saws (as mentioned in my initial suggestion) don't have much conviction as far as stiffness goes until you go upmarket (and you don't want that for what you're doing), thus the spine recommendation. sans spine, it's perfectly acceptable to grab the back of the saw and make it a two handed operation until there's some cut depth. That'll leave a top line that's more crisp (and speed things up). More point and shoot, and less floppy saw plate.