2 solutions:

• enable "droplets only", duplicate base mesh and remove the dropletgen modifier of the second copy.

OR

• in your material editor, add a new "Attribute" node, type in "UVMap" and connect it to the vector socket of your texture.

Make sure you apply the scale of your object (Ctrl + A).

Apply the scale and/or rotation of your object (Ctrl + A)

Make sure you select your weightmap under "Weight Map" option (click on the cross like symbol and select it from the dropdown menu)

Enable "realize instances" before applying the modifier

Disable "Long Droplets" (if you don't need them) or increase its count slightly, increase "Big Droplets" count and/or play slightly with the scale/density under "Distribution"

Enable "realize instances" before you go into Render preview

Enable "realize instances"

Long droplets work on flat surfaces only for now, very experimental, work in progress.

Make sure you don't have inverted normals on your object (in Edit Mode: select all > Shift+N)

Make sure you are using at least blender version 3.1.