Looking at your armory, your gear looks gemmed and enchanted decently enough, maybe not perfectly but certainly not the source of low dps. 1:1 crit:mastery ratio is the current wisdom for secondary stats after your haste breakpoint.
Only having a raid finder weapon will hinder your dps a little bit, but again it's not the source of your low dps.
Your logs reveal the source of the problem, I've looked at Fallen Protectors and noticed a few things.
- Mind blast usage: The encounter lasted 339 seconds. Mind blast has an 8 second cooldown and one second cast time. Given that there is a lot of target swapping on this fight, we'll say that you should do a mind blast roughly every 10 seconds. This means you should have done about 34 mind blasts. You did 24, which is roughly only one every 14 seconds. Mind blast should always be your priority to cast unless you have three orbs which should have been used anyway before mind blast came off cooldown. More mind blasts means more orbs for devouring plague which is a huge source of damage for shadow priests. Make sure that your mindblast cooldown is somewhere easy to see in your UI. There are tonnes of addons for cooldown tracking.
- DoT uptimes: Your VT and SW:P were up 71% and 74% of the time. On all fights but especially multi-target fights you need to aim for a much higher uptime on your DoTs. On multi-target fights like protectors you want to put your DoTs on all of the main targets, not just the one you are focusing. Getting an addon to track your DoTs on multiple targets is pretty much mandatory for this, TidyPlates is great because it shows dots above nameplates, SPTimer is great if you like timer bars.
- Not many shadow word: deaths: It's hard to estimate exactly how many SW:Ds you should cast in a given fight but 6 is definitely low. Look out for low health adds and death them whenever possible since it gives you shadow orbs. Otherwise use it whenever a main target is below 20% every time it's off cooldown.
I recommend checking out the raiding primer thread, it should cover most of these things and plenty more.viewtopic.php?f=19&t=1701