Hi Karen,
Thank you for posting!
Since the connection drops when you are hard-wired into the modem, the issue must lie somewhere between the modem and your ISP. It could be that the modem is failing and needs to be replaced, or it could be the internet signal is weak or getting interrupted.
Reach out to your ISP and let them know this is happening. They should be able to run some tests on your service and equipment and possibly see the drops in their logs that will provide some insight. It could just be that cabling or other hardware needs servicing or replacement.
It could also be interference you and the ISP have little control over. Satellite is known to be susceptible to all sorts of environmental variables, even sunspots!
Hope you are able to sort it out! I can imagine how frustrating that must be.