Personal file-sharing help please

This is probably kind of a strange question, but over my academic/professional "career" thus far and as I am currently applying to grad schools I am often updating Resumes/CV/cover letters, etc. and when I do so I usually just end up emailing them to one or both of my primary email accounts as attachments. I do this because more often than not I am on the go and working on a variety of different computers, including some public.

Besides a thumbdrive (would rather not have to carry it on me) is there a better, more efficient or reliable method for storing/editing documents like this that will allow them to be readily accessible to me and with the newest/most updated version from any computer? (under the assumption that I always have internet access)

I use Dropbox in Windows, IOS and Android.
https://www.dropbox.com
You can use shared folders for what you need.

Or OneDrive. Windows uses OneDrive now, it's nice to have files anywhere you have internet.

I have DropBox and OneDrive but I like OneDrive more.

I have a Windows phone, tablet, and laptop so anything that I have on one is on all. It makes things nice and easy.

Yes OneDrive is handy for Windows devices.
I use it and also Google Docs for my BP record-keeping spreadsheet and a few other things.

http://www.google.ca/docs/about/