When you say continuity checks on the coils did you mean ohms or just continuity? Meaning resistance or simple complete circuit?

You need to test the solenoid coils for actual resistance. The 2 terminal coil should be 1200 to 1300 ohms, the 3 terminal coil is about the same across terminals 1 and 2, and 500 to 600 ohms across 1 and 3... Your meter may autorange to kilo ohms so you'd see 1.2 Kohms instead of 1200. If it varies too far from those numbers it's bad.

The valves shouldn't be buzzing the entire time the ignitor is on, just when it reaches temp and the flame sensor is satisfied. I'm going to assume that is what you mean. If they are buzzing they are being powered; meaning the rest of the system should be happy, and the coils should be opening the valve. So either there's no gas supply or the coils are bad. The internal valve itself is unlikely but also a possibility.

I assume you've verified that the house gas valve is on ? I've seen it be off many times and I've left it off a few times myself and been mystified why everything checks ok but no flame, it's quite embarrassing. I've also seen people use teflon plumbers tape (incorrectly as it is not usually approved for LP or natural gas) and block the fitting or hose with a layer of it.

The coils come as a set of 2. Look for a small metal sleeve that is inside one, it needs to be on the post it was found on and often comes out in the coil. Replace both, it's actually cheaper as a set. It's a whirlpool part number but the exact same coils go in most US designed machines.

Whirlpool 279834 Dryer Gas Valve Coil Kit - AppliancePartsPros.com