Just a list of things I've observed that I feel should be addressed:

1. Purging needs to be a priority. I'm guessing this comes down to two problems: poor positioning and stacked healers on one side.
The former can be fixed easily, we just need to sit down and talk about where everyone needs to stack.
The latter, however, can be fixed either by having more people with purge (not ideal, huge resource drain esp if tanks run it) or by implementing home strat.

2. People should learn the timings of various events in the fight.
For example, once bosses disappear you should already be near the candle line ready to swap, that way swapping takes almost no time. Once I started doing that, I haven't died to prayer phase at all.
Another example is when adds spawn. While I don't know the exact timing, I do know it's about 10-15s after you get your color, which is a few seconds after the bosses are swapped and on the correct side.
Learning these help you reduce the amount of time you're wasting and indirectly will boost DPS, especially on the latter one. Ideally you'll have an eye on your adds and can swap dps to them as soon as they're pulled.

3. On the subject of adds, few things to note from puller perspective.
a) We are limited by pull range, which means for Wills if they're too far on other side, it's not possible to stack them on boss. I try to stack them together on right side ASAP. My question is whether we should do a second pull to stack on boss or leave them to be burned?
The former is great in theory, but with multiple adds, there's a lot more failure points (them getting stunned by something, which has happened a few times) and it burns DPS resources even more by having to reapply DOTs (not to mention potential ults wasted).
b) LOS is a big problem when more adds come. Sometimes adds of the wrong color are blocking, sometimes bosses are. This causes long delays in stacking the adds. When tanking Skin'rai, it might be advantageous to pull him over 5-10m if you have a Will spawn very close.
c) This has been mentioned a lot, but not having any pullers on Vashai side is a near guaranteed wipe. We're essentially gambling with our current strategy that at least one puller will be there every round.
You have a 60% chance of being randomly swapped (5 people on each side, 3 chosen, 3/5=.6). With a puller on each side, that means the chances of each event are:

Neither swap: (1-.6)*(1-.6) = .16 (16% chance)
Both swap: .6 * .6 = .36 (36% chance)
Only one swap: .6 * (1-.6) = .24 (24% chance)

These number change depending on the starting amount of each side (30% chance of having both pullers swapped from Vashai side, 10% chance of neither swapping if both are on Skin'rai side), but having a wipe every 1 in 4 rounds due to this isn't sustainable. I think this is a good reason to switch to home strat even if it's painful in the short term, it will guarantee we have a puller for each round.
This also fixes problems with healers being on same side, alleviating the problem from 1.

Overall, our execution has been getting better every session (even if we have off nights); I feel like these are our main roadblocks from being able to consistently beat this boss.