Eventually, I was able to get him down after taking a break from solo work. After making my spec changes to get Eye of the Storm and Earth's Grasp, the second attempt at Grillok was smooth as butter. The pushback immunity helped take down the mobs faster and the added life to my Stoneclaw Totem made it last a lot longer in keeping the extra baddies at bay.
With Grillok out of the way, the questing in Hellfire felt a lot faster. Partial credit can be made to the spec change, but the larger part of the credit goes to my gear. When I first started out, I had zero elemental gear. All I had left over from the 1-60 grind was my enhancement gear. Having the right spell damage gear makes all the difference world, mobs die faster and I lasted longer in battle. I was enjoying grinding down mobs now instead of trying to avoid all unnecessary mobs. The travel from point A to point B was no longer "how can I avoid as many mobs as possible" to "how can I leave as big of a trail of destruction as possible."
While I was reaping destruction upon the lands of Hellfire, I managed to ding 62! I have proof! Be careful on clicking on said proof though, it's a full size shot of my UI at 1900x1200. My main gaming machine is a 17" laptop hence the wide proportions.

With 62 came the greatest gift to Shaman ever, Water Shield! Any mana issues I had before had swiftly gone out the window. Since I hit 62, I have barely had to drink unless I was killing too many mobs too fast. I LOVE this spell. I wish there was an equivalent for moonkins, but with the new mana regen abilities of moonkins come the expansion, I feel that will be less and less of an issue.
Stay tuned for more about Water Shield and Bit hitting 63!