Sounds like your snare head is either too tight or too heavy, or both.

Snare head is a Remo Ambassador hazy which is a common head. Snare head is tuned to an F#, I don't know if that's too tight. perhaps other snare wires could help? Puresound 16 strand which could fit better in the smaller and deeper snare bead?

When my old Slingerland snare was fitted with wires that were too wide for its snare bed, it acted a little similar to what you're describing. I could actually get the drum to sound decent (with plenty of snare tension), but was having MAJOR problems with sympathetic buzz. Switching to narrower 16-strand PureSound wires fixed my problem.

Before resorting to buying anything, you could examine the snare wires closely to check how they're making contact with the snare head. With the snares engaged at a reasonable tension, tap along the length of each wire with your finger to see if there are places where the wire isn't sitting flush with the head. When I was using wires that were too wide, I found a spot where there was a tiny little gap between the middle wires and the snare head.