For a given number of pixels, a crop sensor will indeed have a larger pixel density, thus a potentially larger resolution. But as the pixels are smaller, you'll have a bit more noise.

The real difference comes when you look at the size of the image on the sensor: you really like to have the subject fill the field of view. If you use a "real" macro lens which gives you a magnification of 1:1 (or more), at 1:1 the image is the same size as the subject. For a 1.5× crop sensor, a subject of 16 mm will take up the full height of the sensor, for a FF sensor, you'd need a 24mm subject. So the apparent magnification on screen (or uncropped print) will be larger for the crop sensor.

While not strictly related to the question, keep in mind that there's more to take into account when setting up for macro: when you are working at or close to 1:1 magnification (or more), handheld is getting tricky. So a good tripod is needed (and one that can go down to ground level). Also focussing becomes difficult, easiest is often to use a rail. Etc... All these extras have to be taken into account for the budget.