Coax outlets are not wired room to room, but room to a central point. Think of your electric service, which has a central breaker panel where all the wall outlet wires join (meet).

An F81 adapter just allows 2 coax cables from 2 rooms to be connected directly together.

How are you converting from the 50ft coax cable to Ethernet?

Sounds like some wires may not have been punched down completely. Gigabit Ethernet requires 4pairs of wires while 100Mbps requires 2 pairs.

If the coax isn’t being used for anything, then that can make things simpler. But, just because you have coax wall outlets, doesn’t mean they are interconnected together. You will need to locate where the coax cables meet, so you can interconnect them together.

If you are only planning two locations to be interconnected with moca (one at router and one at your room), then interconnecting the two cables with a F81 barrel adapter will create the link. Attach the moca devices to coax wall ports, connect Ethernet to router LAN and Ethernet to your switch, and you are ready to go. Enjoy the speed!

Ethernet Switch

https://www.amazon.com/TP-Link-Ethernet-Splitter-Unmanaged-TL-SF1005D/dp/B000FNFSPY/ref=sr_1_11?keywords=ethernet+switch&qid=1701207722&refinements=p_36%3A1253503011&rnid=386442011&s=pc&sr=1-11

This sounds like a bad Ethernet cable. Standard practice is the Ethernet link will negotiate its speed to 100Mpbs (from 1000Mbps) when issues in the link occur.