Here's what I'd do:

1) I'd start by sharing your story on every possible relevant social network in a non-sleazy, non-spammy way. This includes Reddit, Inbound.org, GrowthHackers, and anywhere else you can think of. It's important to understand how those sites work so you approach it in the right way and without coming off as spammy.
2) Next, I'd set up a KingSumo giveaway and offer a liftetime free package to anyone who signs up for the giveaway with extra signups for sharing. Play up the value of getting this service for life, and I'll be the first to sign up. Once you have this set up, reach out to every single internet marketing blog, expert, and Facebook page you can without annoying everyone. Come up with a list of people you think will be interested in the offer and reach out to them. This should get you to 3,000, but if not you can always advertise the giveaway on Facebook, Reddit, Tumblr, and other sites like these (maybe Twitter?). AdWords isn't necessary for this since you'll pay a premium.

I'm pretty sure both of these will get you to 3k while being under $10k, but if not you can always add in...
1) Facebook ads to advertise the free package.
2) Hiring a guest post writer to write posts for you on relevant sites with large audiences. If this is done right, I'm sure you can get to your goal with just this tactic alone so long as you use the right byline to advertise the free package and also target the right sites.
3) Sharing data and lessons is always a good idea. If you have any data/lessons you've gathered/learned from the product you can share those on the blog and then share the post around the web.

That's my quick list. If you focus on a few of these, you shouldn't have any trouble getting to 3k with a $10,000 budget.