Linking the achievement is something you can get around. I've had more than one story of some guy calling for linked achievements and my buddies link that achievement....despite the fact that they haven't achieved it yet....and they get an invite. There are in fact some people who don't actually CHECK achievements. Futhermore if you're more than geared for it and they whine about it, check THEIR achievements and see if they have it. If they don't, you can always mention it in trade right after he spams again looking for people. That either earns you their ire or a real quick raid invite.
Heroics are basically the order of the day, I'm sad to say. You may have the gear to do lower-end content, but people these days basically only have eyes for ICC. Depending on your server it can be near impossible to do anything but that, the weekly, or Ruby Sanctum. MAYBE a ToGC.
Sadly, the only real option is grind your way to absurd amounts of triumph badges. Preferably the...think it's 75 that you'd need for the 245 shoulders, and then do it again for the 245 helm depending on your Tier 9 set bonuses. Make sure to try and do your weekly for the extra triumph and frost, and just keep on pluggin' away at it. It may take you a few weeks up to a month to get yourself geared properly. But you at least have the advantage of having previous tiers of raiding-gear on your side. Some things like trinkets and rings can be skipped if absolutely required. Especially these days when tanking tends to depend on a big health pool over avoidance. (thank you Chill of the Throne)
Bottom line: Heroics....lots and lots of heroics, sadly.
"I don't have a dream, so I protect the dreams of others."~Inui Takumi