I have DDWRT based WiFi router and affected the same bug.
DHCP6 as well as DNS6 in the router are disabled.

My problem was, that local DNS names provided by router stop resolved after few minutes since reconnecting to the router WiFi or LAN, while global domain names seems always resolved fine.

Bypassing systemd-resolve by editing  manually nameservers in the /etc/resolv.conf which is linked to the ../run/systemd/resolve/stub-resolv.conf can fix the problem with local DNS names.

First I tried, as it was suggested in some discussions of the similar issue, without success to set DNSSEC=no explicitly. Seems this systemd-resolve setting is default now.

Then I tried to set DNS servers manually in the Network Manager. This didn't help also. 
Here the output when local DNS names provided by router stop resolved.
---------------8<----------------
$ systemd-resolve --status
Global
       LLMNR setting: no                  
MulticastDNS setting: no                  
  DNSOverTLS setting: no                  
      DNSSEC setting: no                  
    DNSSEC supported: no                  
          DNSSEC NTA: 10.in-addr.arpa     
                      16.172.in-addr.arpa 
                      168.192.in-addr.arpa
                      17.172.in-addr.arpa 
                      18.172.in-addr.arpa 
                      19.172.in-addr.arpa 
                      20.172.in-addr.arpa 
                      21.172.in-addr.arpa 
                      22.172.in-addr.arpa 
                      23.172.in-addr.arpa 
                      24.172.in-addr.arpa 
                      25.172.in-addr.arpa 
                      26.172.in-addr.arpa 
                      27.172.in-addr.arpa 
                      28.172.in-addr.arpa 
                      29.172.in-addr.arpa 
                      30.172.in-addr.arpa 
                      31.172.in-addr.arpa 
                      corp                
                      d.f.ip6.arpa        
                      home                
                      internal            
                      intranet            
                      lan                 
                      local               
                      private             
                      test                

Link 3 (wlp1s0)
      Current Scopes: DNS                     
DefaultRoute setting: yes                     
       LLMNR setting: yes                     
MulticastDNS setting: no                      
  DNSOverTLS setting: no                      
      DNSSEC setting: no                      
    DNSSEC supported: no                      
  Current DNS Server: fe80::22cf:30ff:fece:c98
         DNS Servers: 172.20.1.250            
                      172.20.1.1              
                      8.8.8.8                 
                      fe80::22cf:30ff:fece:c98
          DNS Domain: ~.                      
                      lan                     

Link 2 (enp3s0f0)
      Current Scopes: none
DefaultRoute setting: no  
       LLMNR setting: yes 
MulticastDNS setting: no  
  DNSOverTLS setting: no  
      DNSSEC setting: no  
    DNSSEC supported: no  
---------------8<----------------
Any idea what was the IP v6 DNS server address fe80::22cf:30ff:fece:c98 assigned?

I fixed the problem by changing Network Manager IP v6 mode setting from Automatic to Ignore.