Resident Work Visa while having valid Saudi Work Visa

This is what I know for sure. It would be possible to enter Qatar with the employer visa. You would, however, not be able to re-enter Saudi Arabia as you will be considered an absconder who did not fulfill employer contract. This might be deliberate on the Saudi company
The safest option is to return to your home country and let your KSA permits expire, which will ensure you won't have problems entering another country.
If the job offer has to be taken immediately, you have several options. You can either take risk (risky considering Qatar also has a kafala system) or perhaps check with your nearest Qatari consulate.
Have a look at visas for Qatar and Visas for Saudi Arabia pages to get a little more information.

Dear Wilson,

Thanks for the detailed reply. I have no problem with Saudi Visa status. My problem is that I should not have any problem entering Qatar, expiring re-entry visa before entering Qatar is a good option.

Thanks a lot,

They will deny you in the immigration counter if you have valid GCC visa in your passport. I suggest that you return in your county. Get new passport by claiming it was lost. Then process your Qatar visa under new passport. I have a friend who did the same. It should work if your sponsor did not blacklist you because as per MOI biometrics database is common in GCC countries.