I'm not sure if I'm qualified to answer this, seeing as I'm completely new to wordpress.org. So please forgive me if I'm stating the obvious...
You had a free blog at wordpress.COM, then moved to wordpress.ORG so you could use your own domain, etc? If that is correct...
I have some videos from dailymotion and youtube also. My experience thus far is that posting them is a little quirky, maybe you do need a plugin (?) BUT I have have had success by doing the following:
open your wordpress page or post, and click on the code tab
grab the embed code from youtube (instead of the url one)
paste it into your code tab, then without going back to the visual tab save/publish your entry. For some reason (this has been my experience) clicking back on the visual tab distorts the code so that it doesn't work.
same steps basically, use the embeddable player code from your video, again, don't click on the visual tab, just save it immediately. I have gone back to edit the post and had to re-copy/paste the video codes. Not sure if anyone can elaborate on that, but I just do what works!
I hope this helps you!