YES the 13.6VDC output will re-charge your two new T105 6VDC batteries connected in series however it will take a good 12 hours or more to charge them.
Also with the 13.6VDC constant DC VOLTAGE output from your single mode converter/charger will boil out your battery fluids if left on charge for a long period of time. You will need to keep checking the battery fluids to be sure they have not been boil;ed out.
You really need to replace the converter/charger with a smart mode converter/charger and should be at least a 45-55AMP model for the two T105 batteries. BESTCONVERTERs is a good place to get a replacement model.
The multi-mode smart converter/chargers will really take good care of your T-105 batteries for you. The smart mode converter/chargers can also re-charge your T-105 battery bank when connected to the 120VAC receptacle from a generator in about three hours generator run time when camping off the power grid.
Roy Ken