Starting issue. Starter or battery? Both new

02 Mercury sable. 80k miles. From Florida, now in
Maryland.

So my battery seems fine, I’m stuck 60 miles away from
home and after a nice Valentine’s date with the old lady the
car won’t try to turn over. Headlights work fine and don’t
dim when I try to start, radio and dashboard lights are good
too. Turn the key, no clickclickclick noise, with my head
under the hood I can maybe hear one click, but I think it’s
coming from the ignition when it’s being turned. Doesn’t
sound like anything coming from the starter. Battery
terminals are grounded. I have basic tools in the trunk but
my multimeter is at home.

I drove all around town today with no issues. But it seems
to act up in cold weather. Check engine light came on as I
drove up here last week. Code turned out to be evap leak.
Got new gas cap and all seemed well. We parked, had
dinner, and maybe an hour later came back to find it
wouldn’t start. I don’t have jumper cables and did not get a
jump. (I know I know). But since the lights worked and
didn’t dim as I started it I figured it wasn’t the battery.
Ended up with a local hotel room and am going to try again
in the sun when it warms up.

I got this car from my aunt, it sat in her parking lot for two
years before she gave it to me. She paid a shop in FL to get
it running so they changed fluids, new battery, new starter,
New oil filters. Since having it I’ve changed the thermostat.
And the heat doesn’t work but that wasnt an issue until
come up north. But that’s probably not relevant to the
starting issue.

I’m hoping I can get a jump tomorrow and that or the warm
temp will fix it. Could the starter be bad even though it was
replaced a month ago? Is the solonoid part of the starter?
Should I clean the connections to the battery terminal?
Really don’t want to have to pay to tow it so far.

Thanks for the help, yall