Lithium batteries are a bit more mysterious than your average Nickel or lead based rechargeable. At a certain point it will not be used due to fire risk. The 3.7V is a nominal cell voltage. They are about 4.2V fully charged. The 3.2V is close to flat for a 3.7V nominal cell.
Several models (DXG, KK3 at least) will not turn on if the battery doesn't charge.
3 x AA Alkaline cells gives about 4.8V when new & unused, and the useful endpoint is 3.0V even though the nominal voltage is 1.5V per cell.
The issue is that the Lithium batteries / cells, unlike Lead Acid, Nickel Cadmium or Nickel Metal Hydride (NiMH) will go on fire or explode if you attempt to charge them when they are worn out. So when the device stops charging and refuses to start the cell is still about 3.0V to 3.2V.
The Zinc Carbon end at about 0.9V per 1.5V cell and if left running (torch or motor or radio) the casing will dissolve and acid leak out. The Alkaline cells will also eventually discharge from end point of 1V and then if the spring is too strong the base seal will fail and the alkaline solution will leak out.
You need a new cell.
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