Only one side of 220V Outlet has power. Dryer shuts off after 5 minutes
First, you need to change that to a 4-prong socket (NEMA 14-30) and cord (and I bet the dryer came with a 4-prong cord).   The 3-prong type is obsolete and dangerous.  If the neutral wire loses contact, it will energize the chassis of the dryer - and I bet the washing machine is grounded, isn't it? :)  It's a matter of time before you touch the equipment in the wrong way, and blammo.
How likely is a wire losing contact? It's happening to you right now.
That plug has been retired for decades - it's only sold today as unit replacements for houses which can't use a 4-prong.  You definitely can -- there's a ground wire inside that junction box.
Now, if the ground wire is already tied to the metal box, then you will not need to run a ground wire to the 4-prong socket -- it will pick it up automagically via the hard flush contact with the metal box.

As far as your immediate problem, it's not defective wires. It's unbelievably unlikely for a momentary surge to destroy a wire.  However, it's very common for surges to take out wire connections (where they meet the screw).  We presume the dryer end is good because you just redid it...
... so that leaves the breaker end.  Your test measured voltage to the breaker screw, not the quality of the wire connection at that screw.
Don't forget to check the neutral wire inside the service panel... everbody forgets that one.  Remember, a loose neutral with a 3-prong socket creates the condition that will kill you so treat that dryer with fear and respect until you have either installed a 4-prong outlet, or have positively cleared this problem.