Either of those words are in correct usage; however, the first two phrases would be: I am almost crying. I am nearly crying. (present tense) I almost cried. I nearly cried, (past tense) I ate almost everything but I wasn't even hungry! I nearly ate everything even though I was not hungry! These are correct, but the phrases would sound more natural using 'about to' in American English. For example: "I was about to cry." or "I am about to cry."