My website is designed to meet the accessibility guidelines.
I\'m HOPING that this means screen readers should work well with them... But I have two questions:
Is this a fair assumption to make?
No, even if you think you know the ins and outs of accessibility well, only testing can really tell you this.
Are there any free/cheap screen readers clients I can use to test or online emulators?
The already mentioned NVDA looks like a viable option, or you can download a trial of JAWS which I believe is the most widely used screen reader on Windows. If you're really serious about accessibility and you have a requirement for ongoing testing, you might want to think about just buying a copy.
On a final note, it sounds like you already know this but no amount of automated accessibility testing can really tell you if your site is accessible, only real-world testing can do that! Which is what you're doing, so well done.