I don't know a whole lot about the tri-slosher other than move around a lot and fling paint, but there are a few things I know about the other sloshers.
Sloshing Machine: From my experience trying to use this weapon, yes, use special saver. Also (and more importantly maybe) I find it quite weird/difficult to nail directs with this thing. The mechanics are so much different than any other weapon, but it does mean you can hit people hiding on walls while being on the other side. Basically if you use this, get good at moving around to be hard to hit, while practicing getting directs for 2-shot kills rather than 3. If you decide to use the tri-slosher or sloshing machine you'll focus more slaying (killing people) while the other sloshers focus more on control and keeping enemies away from wherever you want.
Slosher: I prefer the slosher deco so all I can say about this kit is use ink saver main (14 sloshes a tank is quite awful) and I've seen people do well running special charge or sub saver as well. The kit can be used for zoning people out of areas you don't want them to be coming and this comes especially easy with this weapon with suction bombs. Throw a suction bomb and no one is coming that way and then if they do just slosh em. There's something that's really important to know about using the slosher but I'll put it in this next part.
Slosher Deco: It paints. It paints more, and it uses baller to kill people. Sprinklers are great for distracting people and painting and helping to hold down areas (like the zone in splat zones). You can also chuck it into the enemy base so they have to clean it up (else they don't have as much control of their base making them weaker if you push). Basically just keep your distance and you can kill people quite easily. If a mid-range shooter comes close by like a splattershot then try and get in one slosh and follow up with a second one to kill, but if that doesn't work after the 2nd slosh you should swim a little bit around to reposition yourself for a second slosh to kill them while making you harder to hit. If you just stand still sloshing you're an easy target unable to react to anything because of end lag. If you ever find yourself unable to make it out of your side of the map into the middle then that's where you build up baller then go use it to push them back or take zone or get a few kills for your team to get back in.
This is the set I use for slosher. Some people use baller to protect them if they're about to die but I prefer to use it to push. It's really up to you though. If you want to use it to give yourself a little shield when you're about to die then you probably don't need as much special power up but again it's all up to your preferred playstyle.
Here is an advanced slosher guide from Splatoon 1 but I found it to help me out with understanding how to play the weapon better. Hope I could help! :)
