Why can't I cry? I get close, but I just get choked up.
Crying is kind of like an orgasm except you don't feel guilty afterwards. If you are too self-conscious, you'll be tense and your awareness absorbed in your mind rather than in your body. I'm going to assume safely that as you approach the crying-threshhold, you get excited and become aware of the fact that you might start to cry. You might get so excited that you try to draw it out, mentally. That's the point where you're becoming self-conscious and tensing up.
The solution is to give up the self-consciousness and over-thinking. That's tricky, because we're used to accomplishing things through thinking and controlling our thoughts. If you try and apply that to crying, having an orgasm, or anything that requires sponteneity (e.g., humor, creativity, empathy) you're essentially using your thoughts to try and stop your thoughts. That's like forwarding a chain letter that encourages people to take a stand against chain letters.
Forget that. For the next 3 weeks, load up on issues of the heart. Find things you find emotional, beautiful, and pure. Inspirational kids would be a pretty safe bet. Touching music is good too. Things that have themes of hope and renewal. Spend about an hour a day admiring these things. Here's the catch: for the next 3 weeks, you have to resist any urge to cry. If you feel it start to come on, don't let it out and continue whatever it was you were doing. Don't ask why, just keep an open mind and try it out.