The hose controls how firm shifts are. With no hose, it should be shifting hard all the time. I'm not sure of other side effects, but you can at least run a hose from it to the intake manifold. The one on the car I'm working on has it running straight up and into the crook of the upper intake manifold to the left of the throttle body.
If the tranny is slipping, your clutch plates are probably toast. The solenoid could have been replaced by ripping the side of the tranny off, but at this point it looks like you'll need a full rebuild.
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'13 Mustang 3.7V6 - MT82
'00 GP SE (L67 Swap, XP, Headers, 3.25")
'98 GTP, '01 Bonne