Does my skin need to be 100% healed before my next session?

Should I wait for my skin to completely heal before my next electrolysis session? According to my electrologist - yes. But it’s been over a month since my last treatment and some parts of the treated area which were red for like 2 weeks after treatment (I think it was an acne breakout) are still slightly brownish. It might take quite a while before this discolouration completely disappears. Should I wait for it to be gone even if it means waiting for another month or even more?

The slight brown discolouration is likely hyperpigmentation, which is not permanent but can take some time to resolve completely. It is perfectly normal to continue treatments with hyperpigmentation in the treatment area. A month seems to be a long time to allow the skin to heal before continuing. When I first started, I saw my electrologist once every two weeks. I could have gone once a week had there been enough hair to treat. The main concern is that the area isn’t being overtreated and that treatments don’t occur on an area that has already had a severe or abnormal reaction. It doesn’t sound like this is the case for you.

Ultimately, your electrologist is the one performing the electrolysis, so she will decide when it is appropriate to treat you again. I would be surprised if she expects that you can only be treated after a month though.

My skin is pretty sensitive. It was was still over-sensitive and slightly red after 2 weeks.

Scurvy’s response is spot on. What I would add is that any further treatment will speed up the evening out of the hyperpigmentation.
With regards to sensitivity, the first 2-3 treatments always cause more severe reaction, but afterwords your skin “knows the drill” so you will find that you are recovering much faster and often you will notice the rejuvenation effect on the skin - plumping out and glowing radiance.
With regards to overtreatment, your electrologists would know how to space out the treated hairs to avoid that from happening. Usually, 2 weeks wait is enough for the skin to recover before treatment in the same area. Remember, it is different hair that’s going to be treated to the micro-wound will be in a new micro-area, not the exact same spot that was treated the previous time.
I hope this make sense.