tspeed

Hello, I recently bought a vintage (1990's ?) Performance mountain bike and found out that the
FD thumb shifter is not working correctly.
The FD is labeled Shimano Exage 500 and the thumb shifter has two levers: long and short.
When the long lever is pushed in, it is sticky and takes some force and does not stay put.
It goes back to its original position as soon as I take off the pressure on my thumb.
So it seems like it's not engaging as it should and there is no noticeable click.
The right side shifter for the RD works fine as it clicks when the long and short levers are pushed in.

So I wanted to take apart just the plastic housing of the shifter but not sure how to go about it without breaking anything.

I removed the shifter from the handlebar-brake assembly by removing the hex bold from the top.
Then I removed the single screw on the bottom of the shifter housing which allowed some movement of the plastic housing.
That's as far as I got. It seems like the cable barrel adjuster has to be moved away from the housing to allow the housing
to slip down the cable, but I could not get the barrel adjuster away from the housing.
Can someone tell me how to move the barrel adjuster away from the housing please?

I have attached some pictures of the shifter.

Sorry for the long post.