We aren't given a very clear timeline for how long Sarah spent in Beth's apartment before she ended up at the police station, but my impression is that it was not very long. And remember, Sarah was not initially planning to take over Beth's life. Her plan, at first, was to impersonate Beth long enough to get access to the large sum of money Beth had, then skip town with it. As such, she was not concerned with digging into Beth's background all that much; if she were, then her place-of-work would certainly have been near the top of the list of things to find out. But all she really need to do was fool the bankers -- looks, speech patterns, mannerisms, signatures, etc. She spent what little time she had watching home videos of Beth to nail down those things, and didn't get any further than that before Art interrupted her scheme.
As far as clues that should have been "immediately obvious", as you said, her badge and gun wouldn't have been lying around the apartment. She was a detective, so she would probably have had only a single dress uniform, and when Sarah digs through Beth's closet she clearly didn't find it. Perhaps Beth kept it stored away to avoid damaging it, or maybe she had just sent it out for dry cleaning in anticipation of her hearing, who knows.
Finally, you might expect there to be some kind of memorabilia around the apartment that would tip Sarah off as to what Beth did for a living. But it's also possible that Beth didn't like to "bring work home" with her; we don't see much of anything decorating Beth's apartment, and it may simply be that Beth kept her work memorabilia on her desk at work, and reserved her space at home for personal things.