Will multimeters measure amperage higher than what they are rated at?

Add one of these to your project.

https://www.solar-electric.com/mka-100-100.html

It's important that a current shunt has 4 terminals so that the meter reading does not include any resistance in the screws etc. You leave the shunt in circuit and only connect your meter when you take a measurement. Your meter will be set to measure 100mV full scale. You do not have to interrupt operations to take a measurement since the shunt is in circuit all the time.

Only if you add an external shunt.

No. In fact, they usually measure very slightly less than the rating - if a digital meter is rated 10 amps, it actually reads to 9.99 amps. If the current is higher the meter will give an error reading, often this is simply the figure 1. You won't usually damage the meter with a current just a little above its top range. For lower current ranges which use low power shunt resistors, it is more likely that you will blow a fuse or fry the resistor if you exceed the max. current. If you don't have a rough idea of the current before using the meter, set it to the top range first, then if necessary, turn the setting down to the right range.

You can't measure more than 10 amps with this meter or else you will blow the 10 amp protective fuse inside.

Why does it seem like your project is drawing more than 10A if it reads 10A on a 10A scale? Just asking. Usually when meters overload they flash or read 99999 or something to that effect. If they are reading full scale output, they believe it, not sure I would but maybe.

THIS IS THE CHEAP MADE DIGITAL METER, EXPECT TO HAVE AT LEAST 5% ERROR ON ITS DC-AMP METER READING. IF CURRENT MEASUREMENT IS HIGHER THAN 10A, THIS CURRENT METER WOULD NOT SHOW READING OR BLINKING AND DISPLAY ERROR.

If you connect more than 10A to your multimeter then it will get damaged.