Bins on Augusta Street, Oxford Road, The Oval, Conway Road and Vaughan Street were among those piled high with overflowing rubbish, with bags left on the pavement when there was no room.
This led to councillor Louise Emery branding the mess “a health hazard”.
This was worsened as some dog owners stuffed poo bags into already overfilled bins, which the council criticised.
The council said its staff worked “over the weekend” when they were out from 07:00 GMT to 15:00 emptying bins and litter picking across the county.
The authority added: “It’s disappointing when people are not responsible with their waste, and there is no excuse for dropping rubbish on the floor or fly-tipping.
“If a bin is full, people should use the next available bin instead or take their rubbish home with them.”
Mr Carr said: “You can’t cut staff and cut the number of bins available and expect there not to be loads of rubbish.
“There are less people getting rid of the rubbish and less bins. The council has got rid of some dog poo bins. The public are paying more and more and are getting less and less.”