Hi Nikko,
Per your description, when the Active interface from ASA goes down, like G0/0, G0/1 will assume the MAC address from G0/0 and start to pass traffic. This is regarding the Layer 1 redundancy. The traffic destination will be based at ASA routing table. If the routes point to HSRP VIP IP Address, then ASA will passes the traffic through G0/1 with destination VIP HSRP, arriving to Active HSRP Switch. I assuming that SW1 and SW2 are connected themselves into another group of ports.
If you want to change the Active role for HSRP group, you can track the switch interface connected to ASA. This will change the priority level when the interface goes down, forcing the change of HSRP status. You may have to preempt the HSRP group to recover the role when interface back to UP status.
HTH,
Please rate helpful answers.
Alexsandro Reimann.