So, F-connectors are generally sold as twist on or compression. Cable companies use them ALL the time, but... geez, maybe post your location and hit up a few more forums or ask around at your job to see if someone has a f-connector compression tool. I mean, I would swing by and do this for free just to be nice. It's a sub $1 connector and the tool to do a top notch job of putting a f-connector on the end of a RG6 cable.
You can get twist on connectors, but... well, I would do that before investing in a compression tool.
Yes, the adapter you showed would work.
But, this is the piece I was talking about.
https://www.parts-express.com/gold-rca-female-to-f-female-with-nut-hex-type--091-1200
Hopefully that link works.
It replaces what comes in the wall plate. Just double check that it'll work before you pull off what comes on the wall plate, since I'm not totally sure what wall plate you actually got.
The adapter you have shown will work, and I use them when I don't have the wall plate connection I linked to, but it is good to use as few adapters as possible, and behind a wall, things can get ugly if there are adapters back there that jiggle loose. Better to terminate 'F-connector' and convert to RCA at the plate, or on the front side of the plate, not behind the wall.
I don't solder, so I feel your pain there. We'll get you there.