Actually pay them and all of your articles will be include or sometimes use a bulk submission. There's a package that deals with submission in ezinarticles that is really fast.
Just have a link to your website that has a special offer or something with your affiliate link on that page. Or buy a .info site but they might catch that it is still an affiliate link
Another way is to use free classifieds with affiliate link and link the adds in your articles. USFreeads is good for this.
A free blogger blog is all you really need. Just put some content up on the blog (a review, perhaps) and link to it from your article and then link to the product in your review. Maybe instead of a blogger blog, you could use Squidoo. I asked this same question to a mentor of mine and he said "Start your own damn article directory". Hope this helps.
There are a few ways to work around this. 1. Buy a domain and do a domain forwarding to your affiliate site. Info domains are the cheapest. 2. Use a subdomain to redirect to the affiliate site. 3. Use a free blog like blogger or wordpress. They do allow yoursite.blogspot.com