More than 3.2 millions pets were bought by UK household during lockdown, figures from March showed.
Hope Rescue, in Rhondda Cynon Taf, said the number of dogs being dropped off at its rescue centre in Pontyclun was the highest in its 15-year history.
The charity expects the trend to continue for the next two years.
Charity staff said some dog owners had called a dog warden and pretended their own pet is a stray, or taken the dogs directly to a rescue centre claiming they had found it abandoned.
One-year-old Maggie, an old English sheepdog crossed with a golden retriever, was taken in as a stray, but the next day staff saw a recent advert on Gumtree asking for £500 for her.
Sara Rosser, head of welfare at Hope Rescue Centre, said: "We have to take stray dogs and so fake strays are jumping the queue ahead of dogs that really are abandoned.
"It is definitely unprecedented numbers at the moment."
The centre now has 150 strays - more than it has ever had before.
She said: "The rescues are full and then the vets are ringing us saying 'is there any chance you can take them because we're concerned that dog is going to be put to sleep'."
The centre said these were "desperate times" and others like them were at "crisis point".
Centres are at capacity, Ms Rosser said, because of the increase in people who got dogs during lockdown and later realise they cannot look after them as life returns to normal.
She added: "At the moment what we're hearing from all the rescue centres that we work with is that they are also full and that they are under massive pressure."
Dogs arriving at rescue centres post-pandemic are said to have a higher incidence of health or behavioural problems, or both, making them more difficult to rehome.
Often these dogs have no background information on these issues, which lengthens the adoption process.
Hope Rescue said it had received more than 7,000 applications to adopt dogs in 2021, and has had to suspend applications because of the volume.
Often, dogs cannot be transferred to other rescue centres because they have also reached capacity.
Meg Williams, enterprise development manager at Hope Rescue, said: "We think this is going to be lasting for two to three years, maybe even longer.
"The problems are going to continue, not everyone is choosing the right dog for their household."