Tone or Is It Always About Me?

I'm worried about the tone of what I'm writing.  I’ve written everyday for the last five days, which is great, but I'm concerned that my characters sound too self-indulgent.  In two of the stories I worked on this week, they’re told in first person, so they are naturally going to have more "I" in them.  The other one is third person, but still focusing on one character.  In each story, the character has a problem they have to deal with or some sort of conflict.  What do I do?  Do I think about it as I write, or deal with it later?  Is it okay to have a self-indulgent main character?