Hope someone can help you out.
As far as I know, you won't be able to re-direct people to your new URL, unless you keep the old host and re-direct to your new URL. You might want to keep your old host for a month or two for this purpose.
Buy a domain name. If you run into problems with your new host, you can move the domain to a new host. That way you won't be in the same fix that you find yourself in now.
Re-submit your site to the search engines, after you move to a new host. More and more search engines will not index a site on a free server. A lot of them are purging their indexes of free hosted sites anyway.
I've heard good things about www.5dollarhosting.com. I don't have any experience with them. Buyer beware.
Managing your site is usually easily done with FTP software on a pay host. FTP software can be found for free on the web.
Good Luck.
Some options:
1. Buy your own domain name.
2. Try www.webseed.com - it's not a free hosting service but an information portal. You apply and if approved, they register your domain name. Everything is there for you to use their templates to set up your site. The catch is that you have to follow their format. But you can control most of that and you can also control the advertising.
I have two sites there:
Brother vs. Brother: The American Civil War
The Outer Banks of North Carolina Still in development.
3. Try www.suite101.com - similar in nature to Webseed, but there is much more of a Suite101.com presence than you would have at Webseed. I have one Suite101.com site (their limit)
Fatherhood
See how I can shamelessly promote my own stuff while offering you advice? Best of luck.
Spend $8.95 per month to have it hosted. (www.icom.com)
For about 10 bucks a month your problem is solved.
Free service (ad at the bottom - no pop-ups)