The heirlooms are super effective items. They are equivalent to some of the best dungeon drops you can get(or better) and they keep up every time you level.
Spirit only matters in long fights (five minutes or more) such as raid bosses. It shouldn't matter for dungeon instance running. You should manage your in combat/out of combat time so you can drink if your tank is chain pulling. Sometimes it's worth it to allow the tank get out of range so you can finish your drink.
Resist the temptation to heal stupid. If the tank pulls before you're ready, he deserves swift death. If the dps pull before the tank is ready, they deserve swift death. You don't have to kill bad party members, but you don't have to save them either. If you try to save everyone from their mistakes, you will run out of mana.
I put together the efficiency statistics for my 68 holy priest that has a couple of heirlooms and no WoTLK gear (this research ignores crits, and considers chakra:serenity and Inner Fire). You might be able to use this information to choose more efficient heals and better keep up with the group.
Cast Time Heals
Heal 144 mana 2.45 cast 6792 heal 47.1 h/m 58 m/s 2772 h/s
Greater Heal 448 mana 2.45 cast 14504 heal 32.3 h/m 182 m/s 5920 h/s
Flash Heal 448 mana 1.47 cast 10878 heal 24.2 h/m 304 m/s 7400 h/s
Heal is super efficient and should be a go-to choice. Low heals per cast time is the downside.
Flash Heal is super inefficient and should be a last resort (except when it's free).
PW Shield 464 mana 1.47 gcd 10781 absorb 23.2 a/m 315 m/s 7334 a/s
Renew 198 mana 0.97 gcd 1185 + 1974x4 heal 45.8 h/m 204 m/s 9361 h/s
HW Serenity 152 mana 1.47 gcd 8583 heal 56.4 h/m 103 m/s 5838 h/s
Power Word: Shield is super inefficient - last resort. Holy Word: Serenity is the go-to rescue spell. Renew should be kept up on tank, as it can be extended free by any direct heal via Chakra: Serenity.
Binding Heal 410 mana, 1.47 cast 7880x2 heal 38.4 h/m 278 m/s
Prayer of Healing 342 mana, 2.45 cast 3861x5 heal 56.4 h/m 139 m/s
Circle of Healing 243 mana, 1.47 gcd 2687x5 heal 55.2 h/m 165 m/s
Prayer of Mending 266 mana, 1.47 gcd 3285x5 heal 61.7 h/m 180 m/s
These spells would be mana efficient if everyone took damage at the same rate as the tank. These are group rescue spells. Their efficiency is comparable, so use the one that fits your time and aoe shape (you+1 vs all). Slight nod to Prayer of Healing for low mana per second.
Int: 454, Mana: 7600
Spirit: 239, non-casting mana regen: 421 mana / 5 seconds.
Informally observed mana regen rate: ~170 mana per second when not casting.
Tanks at this level typically have ~12k hp.