Dryer connections have 2 hots, and neutral, and often a safety ground.
Anything since 1990? 1996? requires neutral and ground as separate wires.
Dryers do NOT connect "2 hots and ground", if you thought that, that's incorrect unless you're dealing with a Philippine-model or 5-continent model dryer made for 230/400V.
(the Philippine model dryers are typically North American dryers reconfigured to run on 240V-only, by adding a transformer to support 120V loads and using a 240V tumble motor. The Philippine government requires people in the former American-built districts stop using neutral; to harmonize the whole country to Euro-style 230/400V power.)
24A - 1.3A - 22.7A
That's perfectly normal draw for a dryer. The 22.7A is the heating coil, and the 1.3A is the tumble motor and miscellaneous 120V electronics.
13A - 8.5A - 4.5A
Exceedingly improbable on a dryer circuit. 99.9% of dryer circuits are dedicated circuits that serve only the one 30A receptacle. (mind you 30A circuits are allowed to supply as many 30A receptacles as they please; just 99.9% of the time, they do not).
Now, 13A@120V + 4.5A@120V is 2100W, which is a lot of heat - about 1-1/2 hair dryers going full tilt. So the point of loss will be rather warm.
We can cross off the dryer. First, there's nothing in a dryer that could possibly draw 8.5A of imbalance current (tumble motor stalled and pulling LRA???? It would burn up quickly) and nothing that would draw only 4.5A of 240V that wouldn't also burn up from lack of air movement. (I assume you'd hear the blower running?)
So job 1 is chase that circuit and find out what else is on it. Other 30A receptacles are allowed. Sockets of any other amperage are not allowed. Nothing hardwired should be there either (because it with the dryer would overload the circuit - and why isn't that happening, by the way?)
I don't see anything wrong with your ammeter reading methods. Klein is a perfectly respectable vendor, and the "dryer on" readings are right in line.
Also, get rid of that Square D breaker.
It doesn't belong in a GE panel and will burn up the bus. GE breakers only (or Eaton CL type not BR). The correct GE "THQL2130" is $10.
Ditto that Murray breaker at bottom right, again a THQL1120", $5-ish.