Welcome. My name is ***** ***** would be glad to assist.
I have read the information, and based on the situation, it appears that one wire for switch A is connected to the switch B since switch B can change the state of switch A
Since A only has one switch on the light, there are only 2 screws on the switch, one is power and the other is the wire to the light.
These 2 black wires will be from 2 different cables.
The switch on B having 3 screws of course, is your 3 way setup.
On the 3 ways, one 3 way gets power on the Black screw
The other 3 way gets the light wire connected to the black screw
Then the 2 brass screws have the traveler wires in no particular order connected which is how the 3 ways flip flop the power to the light.
That is the key.
So, somehow you have power to the switch A it seems, since you can turn the light off with B.
Verify the wires int he box, the 3 ways should have a common cable with the travelers, usually a 3 wire cable of which 2 are used between the switches, so you can check the switch that was not changed to find those 2 wires and put on the brass screws.
Let me know how it progresses