Since the pump only takes out about half the water before stopping, it
sounds like the float switch either needs to be adjusted or
replaced. It should empty most of the water before turning off.

Second, if you're sure the water is running back in from the drain
line, as opposed to the sump pit refilling with water running in from
perimeter drains, ground water, etc, then the check valve is not
working. But given it's Spring thaw and heavy rain, the pit may just
be refilling with normal ground water.

Since it sounds like this pump is critical and it's age and condition
is unknown, if it were me, I'd probably replace it. And I'd also
install a second backup pump that has a battery backup too, so that if
one pump fails or you lose power for a few hours you won't get
flooded.

Finally, depending on where the water is being discharged, it may be a
code violation to discharge sink and washing machine water with the
sump pump water.