Gearing up for Raids.

Im on Icecrown. Made level 80. Geared myself up in blue items mostly and weapon shield trinket from votes.
I see ICC or TOGC mostly advertised but 5.2 plus gearscore. Me... im around 3700

How do i progress ? I was thinking in farming heroic dungeons to get 5 pieces set level 232 and hope for some off pieces epics to drop meanwhile in heroic dungeons. But even then im not sure i reach 5k gs.

Is there another route ? i am missing something ? I am prot warrior que for dungeons is instant for me. But i cant play more then 3 in a day or else i go insane of boredom. Also life with kids and work.

I know it been asked manny times but im dying to get in bed so didnt looked deep in the forum for simmilar posts.

178-200 ilvl items: RDF, RDF HC and reputation faction vendors around Northrend. Some BoE can be obtained from AH for relatively cheap.
200-232 ilvl items: ToC, FoS, PoS, HoR on normal or HC. Onyxia 10, OS10/25.
232-245 ilvl items: ToC 25, Onyxia 25, ToGC 10 (ToC 10 HC). VoA 10/25 Fire boss.
251-264 ilvl items: ICC 10/25 N, Ruby Sanctum 10. VoA 10/25 Frost boss.
+271 ilvl items: ICC10HC (LK only), ICC25HC, RS25 N/HC.
About VoA: Try to do Fire+Frost runs if you lack the gear.

From EoT gear 232-245 ilvls, get ones that you need. Depending on class T9 2p and/or 4p is recommended.

Mainly you're looking at daily RDF HC for emblems. Try to farm ToC, FoS, PoS for emblems and boss loots. If you're not taken to those due to lower GS, spam RDF HC some more, and check AH for some upgrades.