HVAC Fan Running

Does it have a switch to run the fan all the time?
When I started running the ac last night, I noticed that it had switched from "auto" to "run". Maybe that was because I had the batteries out all day. I didn't notice what the fan was set on before I pulled the batteries. But, like I said, the fan was running on its own yesterday morning, and it has always been on "auto". But it was running on its own.?

Many of these stats use pretty small relays which can become welded contacts, and some use solid state devices like transistors or SCRs and those can fail

One time a service call, complaint was "no cooling." This was a one time customer and we had not installed the system so I was unfamiliar. Before I entered the office building, I checked the 5 OD units and was unsure, because not all were necessarily "calling." But one the suction line was VERY VERY warm, and it was not a heat pump!!! Turns out the office people had bought a bunch of cheap import "setback" stats and one had failed, bring BOTH heat AND cooling on at once!!! The furnace was heating and the AC was "trying" to cool!!! LOLOLOL
That's funny! A losing battle, for sure. Mine isn't that bad, but I do think it's the stat. I figure I'll try that, since my wife has wanted "The Nest" ever since her daughter got one. Happy wife, happy life!
When I started the ac last night, it ran the temp down to the setting, then the fan stayed running (again), when the compressor shut off. I had switched the fan to "auto" (mentioned above), but it didn't care. I cycled the fan switch a few times to see if it had some dirt I could clean off, but it didn't respond. However, before I went to bed, it was working normally, and still is this morning. We have family staying for a few days, and it's behaving. I'll leave well enough alone, until they leave.
All a bit convoluted. Bottom line: I think the stat is bad.