HOLY SMOKES! You have a much bigger problem here than you thought!
Somehow, between your electric utility and whoever wired your house, something didn't get communicated correctly, and the results are frankly, a hot mess. Your utility provides a 120V, 2-wire, single phase service to your house; this is shown both by the voltage readings you are getting at your main breaker, and by the fact that your meter is Form 1S (look below the right end of the long line of text, and it says "FM 1S" on the second of 3 short lines of text there), which is what is used for a 2-wire service like yours. (A normal 3-wire service uses a Form 2S meter, by the way.)
However, whoever wired your house wired it as if they had a normal, 120/240V, 3-wire, split-phase service coming in; we can tell this both from the undersized service neutral coming in (for a 100A service, normally, the neutral will be the same size as the hots, but Code allows it to be smaller than the hots on a split-phase service for reasons that are explained below), and the fact that they made extensive use of multi-wire branch circuits when wiring your house (I count 5, not including your new dryer circuit).
The reason this all is a problem is because in a 3-wire, split-phase setup, service and multi-wire branch circuit neutrals only carry the difference in current between the two opposite legs of the service or circuit. This is useful, because it lets us save wire, and also downsize service and feeder neutrals if large load imbalances are not expected, as normally, single-phase branch circuits are roughly balanced across phases/legs by the layout of North American loadcenters/panelboards, where no two directly adjacent spaces are on the same phase or leg.
However, if you accidentally connect both legs or phases of such a split-phase service or circuit to the same leg, as whoever wired your house did when they connected the service-entrance conductors to the meter base, not only will you not be able to get 240V out of the circuit or service as you expected, but the current flow in the two hots has only one way to get back to where it came from, and that's the single neutral, which is now carrying twice the current. Since neutrals don't have circuit breakers (they're assumed to be protected because their corresponding hot wires are protected), this leads to the neutral wire slow-cooking itself whenever the circuit is heavily loaded, and eventually your house as well!
This gets worse when you have a service that is wired with an undersized neutral, only meant to handle load imbalances. Now, you have a neutral wire that is already too small for 100A, trying to handle up to 200A of current, before the main breaker in your panel will ever notice anything is wrong!
Fixing this will require your utility to help, so call them NOW!
Since this problem is an issue with the service to your house, you will need your utility to intervene, at minimum to pull the meter so that the service wiring can be safely replaced with something that can handle the situation it's been placed in. Furthermore, while the service-entrance conductors and service disconnecting means in your house technically comply with the letter of NEC 230.42(B) and 230.79(C), the installation as a whole does not comply with the NEC. In particular, using a 120/240V, 3-wire distribution panel with a utility-supplied 120V, 2-wire (Form 1S metered) service is forbidden, as if it were allowed, NEC 710.15(C) would then be superfluous:
(C) Single 120-Volt Supply. Stand-alone systems shall be
permitted to supply 120 volts to single-phase, 3-wire,
120/240-volt service equipment or distribution panels where
there are no 240-volt outlets and where there are no multiwire
branch circuits. In all installations, the sum of the ratings of the
power sources shall be less than the rating of the neutral bus in
the service equipment. This equipment shall be marked with
the following words or equivalent:
SINGLE 120-VOLT SUPPLY. DO NOT CONNECT MULTI‐
WIRE BRANCH CIRCUITS!
The warning sign(s) or label(s) shall comply with 110.21(B).
As a result of this, I would be seriously asking the utility to get you on a proper 120/240V, 100A, 3-wire (Form 2S metered) service, as that would simplify things greatly, requiring only a new 10/3 run for the dryer circuit (to handle the full load of your dryer) and some handle-ties in the panel (to make it so the MWBCs have common maintenance shutoffs, as required by Code). If your utility refuses to upgrade your service, not only will you need to switch your dryer back to 120V, you will need to have the house rewired to rid it of the multi-wire branch circuits!