Test Leads

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13 AWG is only slightly more resistance. What kind of amps are you measuring? Best to use the thickest and shortest whilst remaining practical. Alligator clips can introduce some resistance but this is realm of <0.005 ohm.

It is always a question of degree. A thicker shorter lead will be better than a thinner longer lead. If you want to know how much, you must calculate the ohms. Usual you will compare that value to the DMM's internal resistance at the 10A range, i.e. if the DMM has 60mOhm and the wires has 100mOhm, there is a lot of room for improvement. If the wires is less than 20% of the DMM's internal resistance there is little advantage in improving them (One banana plug connection can easily add 5mOhm).

Open up the DMM and solder the leads right to the board. Least resistance possible and you do not have to worry about the ends going bad.
Use a different DMM to test voltage, even the cheap ones seem to get that right.
HTH
Keith