i'm aware that i need to use a 3S balance board, and a constant
current converter to charge them
You need more than just a current source. The charger must limit the voltage to 4.20V per cell (12.6V total), and (if you want to get more than 80% charge into the battery) then continuously reduce the current at that voltage until it reaches ~1/10 the set charge rate, then shut off. If you are using a basic CC/CV (Constant Current / Constant Voltage) converter then it probably won't shut off, so set the 'float' voltage to 4.15V per cell.
To prevent over-discharge you should have a cutoff circuit that disconnects the load when the battery reaches 3.0V per cell (9V total). If the battery gets very low (<3V/cell) it should be charged at a lower rate until the voltage reaches ~3.7V per cell (11.1V total).
is it safe to draw power from the batteries while charging?
Yes, but the battery won't charge if the load current is higher than the charging current. Assuming you have 3.7A at >12.6V available and the load only takes 1A, there is 2.7A free for charging. However the charge current must not exceed the battery's rating, which might be eg. 1.5A. So you could set the charge current to 2.5A, but if the load is turned off then the battery would get 2.5A which is too much. Therefore the charger has to monitor battery current separately from the load, or charge at 1.5A which will only get 0.5A into the battery while the load is on.
and where do i connect the load to?
To the battery through the 'balance' board.
Alternatively you could power the load from the power supply when it is on, and charge the battery at the same time with its output isolated from the load. This has the advantage that the battery can be charged at full rate even when the load is on, but the circuit is more complex.
If the load can work at slightly higher than 12.6V then a simple diode switch-over circuit might suffice. Details are important though. With the charger and load sharing a common ground the charger must monitor current in the battery positive lead, not negative, otherwise load current would upset the charger circuit.
Li-ion batteries can explode if they are overcharged, so don't charge unattended until you are sure that the circuit is working properly.