Time to check all of the connections. Especially the neutrals and grounds.
Lights dimming when a motor starts is classic loose neutral.
If your not comfortable working inside the electrical panel hire some one.

When you say all the connections, do you mean all the connections in
the panel or all the connections that are on the boiler circuit (which
is a separate circuit)?

I can easily pop open the panel after shutting down the breaker outside
the house (the socket has a breaker) and make sure all the grounds and
neutrals are tight. If they are tight, then what should I check next?

By the way, the issue is dimming and not flickering. I suspect the
latter would be worse although I am not sure.

Thanks,

Michael