Speaking as a player of classes that have channeled goodies and non-channeled goodies...
Channeling is a double edged sword, it isn't "Oh you LOS and he gets free damage!" that you make it out to be, maybe in a perfect world for the priest/warlock but its not a perfect world.
Lets consider. Your downsides when you LOS but have said channeled thing on you.
1) You're taking damage still, yes bad.
2) You can't see them to use an interrupt if you free from CC. Also bad.
Now consider the downsides of the person channeling.
1) Standing still.
2) Cannot cast other spells during channel.
3) Channel is range leashed, unlike dots it breaks the moment target leaves the range.
4) Makes a big glowy line that is more effective at drawing fire then the giant arrow of a hunter's mark.
5) Is nearly always the primary spell school for the user, thus if interrupted locks down most all of the useful abilities of caster..
Now since you're talking arenas vs a priest...
1) You're an elemental shaman, grounding totem locks down priests fairly strongly from my old forays into shadowdom (I prefer healing though) as dots don't hit it and the priest has to totally stop moving to cast a mind blast, or throw a SW which is a cooldown.
2) If he's doing that he can't just hit your tremor totem as he's likely not targeting it.
3) If he's frost shocked probably gonna waste a global cooldown to dispel, meaning not immediately hitting your grounding.
4) Thunderthrowing him away before he can get to fear range not a possibility?
5) If said priest is spamming dots at you while running up he's using all of the GCDs (well their two dots, one is cast afterall which why didn't you wind shear it?)
6) If you trinket the priest's fear you're likely gonna be able to play turret wars with him now, which an ele shaman is apt to win.
7) If you're in an arena where the hell is your team during this? Are both teams just like "Oh snap son, shaman versus priest, lets step aside and watch!"
Shadow priests aren't the omg super power houses you seem to be making them out to be no more then a ele shaman is. If you're right next to those oh so weak totems maybe you need to drop them to the side a bit so it isn't so easy to take them out.
Shadow priest damage is easily mitigated by that wonderful thing called dispelling. Since all but ONE of their dots is magic. Your comp sounds more the issue then the actual class...especially if your warrior is having issues violating the priest's face off.
Sadly my times in arenas are lacking once my priest fell out of favor for my love of tanking instances so some of my knowledge might be slightly dated.