No this should not be the case, your trunk links to each of access switches will be blocked depending on how you have prioritised STP.
Ensure that your root bridge vlan priorities and svi HSRP priorities are configured to force L2 and L3 forwarding on the same distribution switch.
However, if you want to utilise both trunk link to the access switches, then prioritise each VLAN and HSRP priority accordingly. For example
DS1 ------- V1, V2, V3, V4 ----- DS2
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Trunk Vlans 1-4 to AS Trunk Vlans 1-4 to AS
STP priority on DS1 for V1, and V2 4096
STP priority on DS2 for V1, and V2 8192
STP priority on DS1 for V3, and V4 8192
STP priority on DS1 for V3, and V4 4096
This will force the root bridge on DS1 for Vlan 1 and Vlan 2, and DS2 will be the root bridge for vlan 3 and vlan 4. Each DS with be the alternate path for the other vlans.
For each SVI configure the HSRP priority to be the highest priority for each VLAN the DS is the designated root bridge.
Therefore, interface vlan 1 and 2 should be configured with a priority of 105 on DS1 and 100 on DS2 for the same vlans.
Interface vlan 3 and 4 on DS2 should then be configured to 105, and 100 on DS1. This will elect DS1 as the primary for V1, and V2, and DS2 will be the primary for V3, and V4.
