Hi !
Be sure preferences/NewQSO/Enable auto search on HamQth/QRZ is checked
Be sure that preferences/Callbook support has right selection (HamQTH or QRZ) checked and that username and password corresponds that selection.
Prefer passwords that have only up and low case letters and numbers. Avoid special characters like #%/() etc...

If it does not still work you can peek what happens with internet connection.
Start cqrlog. Do not log any qso yet.
Open console and give command:

sudo tcpdump -X | grep HTTP:

Your password is asked before it goes on. (If you do not have tcpdump you can load it with package manager)

Move to new qso and enter a call sign that exist but you have not worked before. Press then tab to move to RST.
At that point you should see at console window something like:

dropped privs to tcpdump
tcpdump: verbose output suppressed, use -v or -vv for full protocol decode
listening on wlp2s0, link-type EN10MB (Ethernet), capture size 262144 bytes

09:49:54.674319 IP hamtpad.37404 > hamqth.com.http: Flags [P.], seq 1:168, ack 1, win 502, options [nop,nop,TS val 3375307540 ecr 1513513305], length 167: HTTP: GET /xml.php?u=oh1kh&p=PASSWD_HERE&prg=cqrlog HTTP/1.0

09:49:54.806812 IP hamqth.com.http > hamtpad.37404: Flags [P.], seq 1:847, ack 168, win 235, options [nop,nop,TS val 1513513339 ecr 3375307540], length 846: HTTP: HTTP/1.1 200 OK

09:49:54.855747 IP hamtpad.37406 > hamqth.com.http: Flags [P.], seq 1:206, ack 1, win 502, options [nop,nop,TS val 3375307722 ecr 1513513351], length 205: HTTP: GET /xml.php?id=c54aeb7512f03394b1c39bde5c92ebcc57c6ea87&callsign=OH2BH&prg=CQRLOG HTTP/1.0

09:49:54.921046 IP hamqth.com.http > hamtpad.37406: Flags [.], seq 1:1389, ack 206, win 235, options [nop,nop,TS val 1513513367 ecr 3375307722], length 1388: HTTP: HTTP/1.1 200 OK

First block you will see your call and password
Next server responds "200 OK"
Then you see the call you typed to new qso with your session id
Then server responds "200 OK"

This is valid with HamQTH. QRZ produces a bit different output.
If you cannot get session "id" for request it means your username/password is not correct.

If you get those 2 "HTTP: GET" requests and 2 "200 OK" but there is no information coming to NewQSO fields (and you are sure that that callsign has entry in HamQTH!) then there is something wrong with your cqrlog.
Then try to create a new log and set all preferences ok for that and check if HamQTh then works.
If so, export all qsos from previous log with full adif export and import them to new log.