Taking to social media on Thursday, the Ambulance Employees Association said one crew was sent to three "severely unwell" COVID-19 patients overnight.
It said two patients were ultimately transported in one ambulance while another crew eventually arrived to take the third patient.
"Treating paramedics almost ran out of oxygen, they ran out of medications," the association said.
"They were calling for intensive care back-up with none available. Our healthcare system is not coping."
The issues came after six days of record virus cases in SA including 105 on Monday, 154 on Tuesday and 198 on Wednesday.
That put the state over the threshold for an official COVID-19 outbreak based on local modelling released in early November.
The modelling predicted a 27 per cent chance of an outbreak, which was defined as "averaging more than 100 cases per day over any three-day period" after borders opened on November 23.
Premier Steven Marshall said while the number of Delta and Omicron infections was higher than predicted, the rate of hospitalisations remained "much lower".
Also on Thursday, SA Health reported shorter wait times at most COVID-19 testing centres after delays for a swab blew-out earlier this week.
It said wait times ranged from less than one hour at two sites to between one and three hours at most others.
Only one SA Pathology site, at suburban Hampstead, was listed at more than four hours.
"Please be patient with staff at testing sites who are working hard to manage traffic, organise logistics and collect samples," it said.
The long lines have also caused delays in test turnaround times, although results within 24 hours are still expected.
