You will have to apply as the main applicant (because of your degree) with your partner as a secondary applicant - 1 year's experience may be enough though more is better. You probably need 3 years for government sponsorship though not in all cases. The cousin can only sponsor you for a regional visa and it only means that you need fewer points on the points test - it won't speed the process.
Australia's visa processing priorities are:
1. Employer sponsored visas - a few days to a few weeks from lodgement.
2. State or territory sponsored visas - a few weeks to get the sponsorship and then a few more after lodgement for the visa.
3. Visas where the main applicant has qualifications and experience in an occupation on the Critical Skills List (CSL) - 3-6 months from lodgement.
4. Visas where the main applicant has qualifications and experience in an occupation on the Migrant Occupations in Demand List (MODL) - none being processed until at least June and then a large backlog processed in order of lodgement. Best guess, 12 months or more for new applications.
5. All other applications - unlikely to be processed for a VERY long time.
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