How low can you offer? Depends on the seller's motivation and amount of equity in the property.
I understand your concern: It's a pretty big gap between, say, $77,000 and $67,000. And even bigger if you go up to, say, $83,000.
You say "People keep having me look . . . " Ideally, you should be working with one agent--one you select. Not a listing agent who may have listed one of the higher-priced homes. First, make clear what you've been approved for. If all you then get are still homes that might be outside of your price range, ask: "Print me a list of all available homes in __________ [location, or x miles from a central spot] that are priced between $60,000 and $70,000." Then see if there are any in that price range that might work for you. If not, certainly consider ones listed for more. But the higher you go--and in your case I'd guess much above maybe $72,000 or so--the less likely you are to be able to get one for $67,000 or less. Again, certainly not impossible, but less likely.
Hope that helps.