Conversion copywriter promoting non-scuzzy writing that sells. Copy Hackers and Airstory. Speaker @ CTA Conf, Microconf, Business of Software, Copyblogger Rainmaker. Canadian do-gooder.
Kudos to you for seeing the value in great copy. I love that you mentioned 37signals, which is an organization that's made copywriting part of almost everyone's jobs (or so they've shared on their blog). MailChimp and Zendesk are two others that people often point to re: great copy that builds a brand and differentiates; Groupon is another awesome example of really, really tonal copy that people actually read (which is more than half the battle).
MailChimp has in-house copywriters, including Kate Kiefer (https://twitter.com/katekiefer), and so does Groupon. I'm not sure who writes for Dropbox or Zendesk, though searching companies on LinkedIn can often reveal little-known in-house geniuses.
The startups you mention have a certain style and tone that I have to say is different from what you'll normally get with a "direct response" copywriter, though by all means check out the link David Berman submitted to you because you never know. I recommend that, to achieve the slightly funky, funny-ish copy you're looking for, you seek out a conversion-focused copywriter with a creative and UX background. You need someone who's totally at ease adopting a new voice / tone and using it appropriately across your site and in your emails; less experienced copywriters might be heavy-handed with the tone, which often gets in the way of the user experience (e.g., button copy that's tonal can lead to confusion). Be careful, of course, not to push your writer to be exceptionally creative -- because a little touch of tone goes a loooong way for busy, scanning eyes.
Here are some great freelance copywriters you could consider: http://copyhackers.com/freelance-copywriters-for-hire/
The link to Neville's Kopywriting peeps is also great.
Before hiring, ask to see a portfolio or get a) links to websites they've written and b) a zip of emails they've written; if a writer is accepting clients, they'll usually showcase their work on their website. Check out their blog and tweets to see if their voice comes through in their own writing. Don't hire bloggers or content creators for a job a copywriter should do. Don't hire print copywriters for web work unless they do both. And when you find a great copywriter, trust them... and don't let them go - because 10 bucks says, they're in demand or about to be.