Hi Milo
No problem it’s a confusing thing to describe,.. I understand now,.. want you want is the deformed object, exported. not the original shape.
The option to “Export mesh with morphs and skinning” is actually a good example of how hard it can be to describe the process.
this option works on “figure"s, and allows you to export the Posed figure, in that current pose,. so , effectively the deformed mesh instead of the original.
The problem is,... that it only works for Figures,.. or objects with a modifier applied to them.
If you Add a bone or two,. to your Model, (which shouldn’t effect any morphs) you should be able to adjust the Morphs to get the shape you want,. then export as OBJ,. using that (export with morphs and skinning) option.
The other option,.. you can export an object which has a Modifier applied to it, like a “Bend and Twist”,. and that “deformed” object will be exported in that current shape,. plus, you don’t need to add bones, and you don’t need to have adjusted any settings on the modifier, as long as it’s applied to the object. you should be good to go.
You can dial in the morph you want,. and export as OBJ, using the option to export with morphs and skinning,
Quick pix :
Here’s a morphing candle, ...the original model, and the re-imported obj;‘s of the exported morph states.
hopefully you can see the modifier tab, which is set at 0.
Hope it helps 
Click thumbnail to see full-size image