I did a quick Google search for that stick as I was not familiar with it and can only conclude that you have one of two problems but either one would necessitate taking the base off the stick at the very least and possibly some more dismantling as required. If you are happy and confident about taking things apart and putting them back together I would suggest the following:
1) The contacts on the slide feature could be dirty/oxidized through use/age and need cleaning, There are several options for doing this including cleaning with isopropyl alcohol (rubbing alcohol @ 99% pure) on a cotton bud, using an electronics safe switch cleaner (be careful as the wrong type could melt plastic parts!) or, if you can get to the slide contacts easily, rubbing them with a pencil eraser to wipe off the oxidation/dirt.
2) A wire may have broken from or lost contact with the slide parts. This could require either re-establishing the contact (i.e. disconnecting/re-connecting a push on connector to clean the contacts) or soldering a broken wire back into place.
It sounds like you have an age/wear related problem to me but without seeing the parts it is difficult to diagnose the issue from afar. It could be an easy fix or it could be terminal but without opening it up you will never know.
I had a similar problem with my X56 HOTAS stick a couple of weeks ago. I lost all button functions on the stick but the axes worked fine. After opening up the base and finding nothing wrong I eventually took the handle off the base (it is removable to allow changing tension springs) and noticed 10-12 spring loaded contact pins in each half (base and handle). I cleaned all the contacts with isopropyl alcohol and a cotton bud and re-assembled and it has worked fine ever since.
I wish you luck with whatever you decide to do.
