There shouldn't be any default GW under vpn-adapter interface. To see what traffic is tunneled, do the route print command. By default (if no spit tunnel is configured), all the traffic is tunneled. So don't bother for the traffic flow from vpn-clients towards your LAN, but pay more attention to the reverse direction (from LAN towards vpn-clients), specifically check that:
- nat exemption (NAT 0) rules are configured for all VLANs subnets for traffic, destined towards vpn-clients pool (you've got those for local lan, so do the same for others VLANs).
- check that there's correct routing information so those VLANs know how to reach vpn-clients subnet through ASAs inside interface.