Mesh not snapping to curve

So I am try to make a seam, but when I give the mesh the curve modifier and apply the rotation, scale, and location to it, the mesh is not connected to my curve at all.

There was no active Curve modifier in your file so i had to add one.
Also, it is not clear to me what exactly you refer to as a “seam”… so i improvised a bit ;).

Make sure you place all object Origins in 0,0,0. Place 3d cursor at Curve’s first point in edit mode, in Object mode change Curve’s origin to 3d cursor before. Now you can add Curve modifier and there is no need to race for a runaway objects. Curve can be manipulated in Object mode by assigning Hooks at points in edit mode (Ctrl-H, Add Hook as a New Object).

All the jazz needs to be parented to one Object if you want to move thing around. Use another Empty created at 0,0,0, select Curve, shift select Empty and Ctrl-P to parent. Same goes to other objects Curve deforms.

Name things proper in Outliner; helps others at least ;).

Awesome! Thanks for your help!