First playtrough. Should I continue the following:

After making it to the bottom of Blighttown, I got punished real bad for a mistake. Forgot to take bloody repair powders. (and blooming purple moss)

Longsword +3 at risk (oh noes. a longsword!11!).

Scrolled trough my weaponry to see if there was something that I could continue with.

Noticed a Greataxe.

Currently two-handing it, trying to kill Quelaag as of right now.

On to the Question:

Should or should I not take this Greataxe with me until the end of the game, ascending it and whatnot, while leveling my strength more so I can 1-hand it and use some parry shield on my other?

Please take in account that this is a first playtrough. I noticed that this axe is rather slow and im unsure if that'll handicap me too much to be able to complete it as a filthy casul.

Praise the toxic \[T]/