Should I just cut them all of? I cut the part that is mushy and leave the rest on the cutting. Did I do the right thing? The cuttings have already rooted

mushy as in turning dark and looks like rotting mess? if that's the case, cut completely off the rotting part. possibly too much moisture and you need to air out enough that new growth will not be wet all the time.

Yes Pete. It turned black and rotting away. It is very hot in Malaysia right now during the daytime and cool at night, so I brought all my cuttings inside the house. Will the buds grow again?

I have cut mushy sections out, over & over until the last node standing & finally rooted. Common chore for me
if I run into mushy sections while rooting in spaghnum moss. In fact almost every yr I run into similar challenges.

I sanitize the knife or sharp bladek before & after usage. Sometimes I seal the cuts with wax.

Paul, the nodes has started to produce young leaves. And these turned mushy and the leaves dropped. Will it produce new buds if I cut it off totally?

Good chance they may not as much energy have been used. Be hopeful for now. If I can help, drop me
an email. I will be sending my brother some stuffs soon & I can piggy back a small gift. Actually once droopy leaves
is sighted on new starts coupled with mushy-ness, it becomes difficult.

pix would help.. but there's always chance :) usually mine rots due to high humidity inside of the humidity bin, specially when the bin is packed to max capacity. at certain point, i separate my cups according to their growth rate. things that grow well in one bin, others in other. and ones that are going well and have branches that are growing, i make sure to air them out little longer so the leaves and the buds will stay dry and not rot. 

more fresh air, and make sure moisture doesn't collect around the bud.  

like paul, i cut them until the rotted part is gone, sometimes over and over. but usually, if it will live, it will put on new bud at lower node and put on new bud and branch. i have one cutting that i put into cup upside down. that thing is having hard time but it's still coming back and not dead yet.