Slow speed weave

Tell you what.... let's don't do slow weave at all. Let's just ride the bike like it was intended and don't worry about going slow and weaving. The only time one needs to weave is avoiding road kill or something that may puncture a tire laying in the road and most of the time that doesn't quaulify as going slow.
You know every morning before I go to work I slow weave.... or when I get off work I slow weave.... or when I go any where I always slow weave.... NOT!

The OP isn't even talking about slow weave like around cones.... he's talking about a problem he's have with his bike apparently. I bet it might be a bad tire. Tire tech I was talking too at the motorcycle shop has said he seen a lot of new tires come in that are bad. Yep new tires that are bad. He said Avons are the worst.

The only time one needs to weave is avoiding road kill or something that may puncture a tire laying in the road and most of the time that doesn't quaulify as going slow

Now you are talking about swerving a different subject altogether.

Weave : to direct or move along in a winding or zigzag course; move from side to side, especially to avoid obstruction

Swerve : to turn aside abruptly in movement or direction; deviate suddenly from the straight or direct course

Ahhh..... they are almost the same. You knew what I was talking about so that's all that matters.

Going slow on any bike puts the emphasis on your own sense of balance. The bike is very well balanced but it has mass. Which means that if you don't balance well, you'll probably find yourself lurching left and right to try to keep the bike upright and going straight when you're creeping ahead at minimum speed.

It's a skill set that needs practice. My slow speed performance is always much better towards the end of the season than the start, I usually come to a complete stop and then calmly put a foot down, the bike is balanced and everything is under control. But, at the start of the season after months off the bike, it's more of a "oh crap quick foot down and brace the bike" type of thing.

And the real challenge for me is the stop on ground with an incline, and more challenging, different grades between the front and rear wheels (such as is often found at intersections and one is turning onto the cross street), and most humans do not have the ability to sense 32 ft/s-squared acceleration due to gravity acting equally on the rider and the bike as they (we) start to tip over, perhaps focused on traffic and such. We all know that sensation of recognizing too late the bike has gone past our ability to stop its appointment with the pavement, and we step off to consider our next move (and check to see who's watching).

I find the less forgiving a bike is to my inattention, the more I have to wiggle the bike from side to side - a concept borrowed from fire control radar which cannot "see" a target because of the math involved and the physics, so the designer induces error to either side of the target, and the combined errors provide (hopefully) sufficient data to estimate with useful precision the target location, direction, and velocity.

The relatively quick side to side motion allows me to sense in my legs the differential effort required to move the bike in the opposite direction - not to be confused with the effect of being on an incline where one leg has to reach farther.

Like @cr0ft , I find this is easier with more experience each season.