I guess one of the typical solutions would be to configure the server port to their own Vlan on the switch and configure a Trunk interface between the switch and the ASA. And in this case there would naturally be no L3 interface for these Vlans on the L3 switch or if there is it would not be the gateway for the actual subnets (gateway on the ASA)
As your current link between the ASA and the L3 switch is a routed link you could either configure a new physical link between the switch and the ASA or modify the current link to a Trunk. If you configured the current link as Trunk then naturally every subnets gateway should be moved to the ASA and the switch would mainly act a normal L2 switch.
Naturally if you want to keep the switch L3 for all the subnets and still route the traffic through the ASA then you could check if the switch supports VRF Lite (or whatever the official term is) and
Configure a VRF for each different network segment
Separate VRFs for the required DMZs
Single one for the LANs unless you want to separate the LAN traffic also (if you even have multiple LAN subnets)
and so on
Using the VRFs would separate the routing of the subnets configured under the Vlan interfaces and you could route all traffic through the ASA. The link between the ASA could be Trunk in this case. Naturally if there is need you could use a couple of physical interfaces on ASA/Switch and configure a Port-channel between the ASA and the L3 switch if you feel that one 1Gbps link is not enough. Though I guess the ASA models throughput was pretty close to 1Gbps anyway.
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