History of that dangerous socket
When they started grounding things in the 1960s, the appliance industry asked for a "pass" on dryers and ranges connecting via NEMA 10 receps (hot-hot-neutral). They were afraid a compulsory conversion to the modern NEMA 14 series (hot-hot-neutral-ground) would result in customer confusion and lost appliance sales. So they haggled with the NFPA, saying "let us 'ground' the appliance chassis to the neutral wire. Yes, that will electrify the chassis in the event of a neutral wire failure, but those are rare, because that wiring is rarely disturbed". NFPA accepted that, provided the appliance makers start designing ranges and dryers to be able to connect to both NEMA 10 and NEMA 14.
Code historically required NEMA 10 dryer and range circuits either use /3 cable (which has a white insulated neutral) or SE cable (which has a bare meshed neutral wrapped around the other conductors). /3 cable started being made with a ground wire, which was awesome, because even though many sockets were going in as NEMA 10, ground was present in the box and the homeowner could convert to NEMA 14 at leisure. However, some installers illegally used /2 cable, abusing the ground wire as neutral. This bad work is not grandfathered because it was illegal at time of installation, and must be removed on sight.
In 1990-ish, NFPA finally outlawed NEMA 10, requiring NEMA 14 going forward. By this time the appliance industry had gotten appliances "ready" for both styles of connections, either properly grounding, or doing the "sanctioned bootleg" as described in the installation instructions.
The problem
The symptom you describe is very typical of a neutral wire (that wire is neutral) that has been disconnected at both ends. The dryer is unplugged, and either at the socket or the panel, the wire is loose and has "opened", possibly with some arcing and sparking first. Since it was heard locally at the dryer, it may be in the socket. Ideally you should read 0 volts to that wire, however due to "capacitive coupling" you read a little bit of voltage. And that phantom voltage sort of "follows the meter around", which is why you get 16V from hot L1 and 16V from hot L2 (which should be impossible, right?)
Very high impedance DVMs (read: most cheap meters) are particularly prone to picking up "phantom voltage". There is voltage there, but possible current is so tiny that it's incapable of even moving the meter movement on an analog meter.
That lines up with the fact that you had the panel changed. A problem at the panel would make sense.
While you're in here, deal with that socket
Modern society isn't safe by accident. A lot of work was done to make society safer. But that's only true if things have been modernized! NEMA 10 has not.
So, there are several paths to getting rid of the NEMA 10 socket and convert to NEMA 14.
If the wire is illegal (solid bare neutral), replace the entire cable run.
See if there's already a ground wire or metal conduit. Often that is lurking unused behind the NEMA 10.
Retrofit a ground wire. That ground wire needs to be #10 and needn't go back to the panel; it can go to any other junction box that has a #10 back to the panel (air conditioner; water heater), or to any box with metal conduit back to the panel, or to any part of the Grounding Electrode System (the bare copper wires going from panel to water pipe or ground rods).
Just don't hook up ground. Mark the cover plate GFCI Protected. No Equipment Ground. Then, fit a GFCI breaker in the panel.
