As long as the 3V3 regulator that you are using is rated to withstand the maximum voltage that the adaptor outputs then from no load up to any value <= Iout_max the adaptor itself should be very fine.
The 3V3 regulator will dissipate power of
Power = V x I = (Vin - 3v3) x Iin.
If Vout is a steady 5 Volts then at 350 MA the regulator will dissipate
(5V - 3.3V) x 0.35A ~= 0.6 Watts.
You need to ensure that there is enough heatsinking of the 3V3 regulator to handle the maximum dissipation experienced. Modest heatsinking will usually be enough to handle that level of dissipation.
A phone charger may have constant current mode when overloaded.
An electronic supply, which is what you have, based on it being very light weight, will be regulated, so that Vout ~+ Vrated, across most of the load range.
At 0% to 100% of rated load it should run well.
If you load it very heavily it will dissipate up to all energy internally or about 5+V x 350 mA or say up to 2 Watt. Most electronic supplies have overload protection and will progressively shut down if they get too hot.
In the worst realistic case , if very heavily overloaded, Vout will drop below nominal and you MAY be able to make it catch fire but most manufacturers try to not let that happen.