A: Mom, I need a new laptop. Mine’s getting old.
B: Not now, Charlie. Judy’s having a moment.
A: What, so my needs come last?
B: Do you know how much a new laptop costs? I can’t even swing the mortgage this month.
Question: Does “swing the mortgage” mean “pay the mortgage” in this situation? Is it slang?
Yes the unsubtle version is “I can’t even pay the mortgage this month”.
The subtle version is “I can’t even find money from wherever place even by doing sketchy shit or working my ass to the bone”.
I tried to find a proper definition of this usage for you, and I found that there are more meanings for “swing” than I had imagined. This definition is close to the usage in your example, which is very informal
Definition of swing it
informal: to do or manage something successfully
If he can swing it, he’ll visit next month.