I don't think there is a way to do what you want in a sketch. I can think of one possible workaround, if you can live with only being able to move your baskets less than the 6" spacing. Split your arcs and straight lines into individiual segments. Dimensions a straight section leading into an arc as less than 6" and then use an equation to drive arc length of the first section of the arc following the straight section as 6"-the dimension of the straight segment. Now dimension all of the other arc lengths and straight lengths to be 6" until you get to the end of the arc where you use a driven dimension that allows you to use an equation to drive the next straight segment.
It might actually be easier to solve this with bodies instead of a sketch. Make a body to represent your basket and use a Curve Driven Pattern to place copies around the path. I think that Curve Driven Patterns use the distance along the curve to establish the spacing.
Welcome to the forum.
Take a look at Insert>pattern/mirror>curve driven pattern
Also take a look at these threads