Charging and Using a Lithium Battery
Generally speaking, safety of Lithium batteries have to be verified on the product and charger is only a part of the product.

Your circuit from Adafruit is based on MCP73831/2. This linear charger IC has the usual constant current/constant voltage charge control and, once termination is detected, it will restart charge only when battery voltage decreases under a certain threshold. This kind of regulation, under some circumnstances, works even if the battery has some load applied.

However, MCP73832 has a "charge termination" feature. Charge is terminated when the system current is below a certain percentage of the programmed charge current. If your system drains constant load from the battery, this current is always exceeded, and that's why charge termination is never detected.

I suggest you to have a read at AN1149 here: http://ww1.microchip.com/downloads/en/AppNotes/01149c.pdf
This application note explains the problem and suggests you some possible solutions. It should be valid also if it is not strictly related to the same charge controller.