My last batch of American brown had an OG of 1.52 and fermented down to 1.020. I had the same worry you did so let it sit on the cake for a full 3 weeks, swirled it at 10 days to try and wake up the yeasties, etc, FG still was 1.020 and stable I guess since it was 11 days between readings.
I went ahead and bottled, I'll put one in the fridge on Sat and give it a few days in the fridge and crack it open to see how it tastse. I have a feeling it is going to be sweet since it stopped at 1.020 (I think the yeasties went to sleep one night when it went into the low 50's in my house, but I tried waking them up and let it sit for over a week after that....)
If it is sweet, I'll just call it a Sweet Ass Brown.
Maybe next time it will work differently, but it is fun learning, when learning means having to drink 2 cases of HB, to fully understand the beer you brewed's character. So I would say just bottle it if it is stable @ 1.020 and see what it is like, if no good, no matter what just relax, tell yourself your learning with each sip, gulp & chug.