Why do the double VAXXED still carry and transmit infections?

The VAX triggers production of the virus spike protein which triggers production of Immunoglobulin antibodies, type G and A.  More G than A and the A begins to decline quickly.

Type G mainly circulates on patrol in the blood but Type A is mainly patrolling the mucosal lining of nose and throat.  Unfortunately, the nose and throat surfaces are the landing zone and first nursery of the Covid-19 virus.  Therefore, the "fully vaccinated" can harbour and breed Covid-19 virus in the landing zone because the first line of defence weakens soon after the shots, even while having a second line of defence circulating in their blood.
Source at NIH

The illustration from the NIH source shows the pinkish area is the main domain of IgG and the lining of inside passages through us (nose, throat,lungs, bowels) are predominantly patrolled by IgA.