You stop the clocks by "turning them off". Here for example I stop timers 1 and 2.
// reset Timer 1 and Timer 2
TCCR1A = 0;
TCCR1B = 0;
TCCR2A = 0;
TCCR2B = 0;
As for the ammeter, yes that is a well-known problem. It is called "burden voltage". Basically the ammeter works by drawing current through a current-sensing resistor, which drops the voltage a bit, and it measures that drop. With a bit of maths (knowing the resistor, and the amount of voltage over the resistor) it can work out the current.
However the voltage drop may actually change the behaviour of the circuit under test. Read more here:
A different meter might help, or make up your own test if possible with a low-value resistor. You could buy his uCurrent device but they tend to be out of stock (as they are today).
Another alternative for you might be to measure the voltage drop yourself, and if it is (say 0.5V) increase the voltage to your circuit by that amount (while you are measuring current) to compensate.