Too much weird stuff going on lately to not be super careful. We're only paranoid if they AREN'T out to get us, but we know that they are!
I don't think Dorothy overreacted at all. I also would have left. The gent should have started the conversation off with, "Hey, there's a cop here being nosy, let's delay a while..." or something to that effect.
I had a similar issue come up recently, although I don't use a PI, but reserve mainstream accommodations. A nervous newbie pulled in and saw what looked like a police vehicle parked in the far corner opposite my room, someone just sitting in it, a la stakeout. He, of course, bolted, but was polite enough to call me from down the road.
I got dressed, grabbed my ice bucket, and nonchalantly strolled outside on the way to the ice machine. Turned out it was a decommissioned PD car (they are often sold at auction with emblems removed but original ownership still quite obvious), guy in the driver's seat on a cell having a smoke (not allowed in the motel) and arguing with what sounded like a girlfriend over the phone. From a distance it WOULD have looked suspicious.
I called my fellow back and offered to meet him at a nearby coffee shop, as I knew he had driven a long way. He decided to come back to the motel, parked next to the car, which was by then empty, we met in the public picnic area, chatted a bit, retired to my room, had a drink and a jolly good time.
Stuff happens. Use your best judgement and be prepared to roll with the punches. -- Fancy, from her phone, where it isn't easy being green