Since at the Event Horizon, time stops completely, how do two black holes merge together? Shouldn't they should stop moving due to time dilation when they get closer to each other's Schwarzschild radius?
The "event horizon" is defined as the point (or surface) from within which light rays can never (ever) reach a distant observer. To find the location of the event horizon implies that you must know everything about the future of the black hole - so in practice what is referred to is often the event horizon of a Schwarzschild black hole, which is static and eternal (or a Kerr black hole if it is spinning). i.e. It never changes and can be calculated.
When black holes merge, they cannot be considered as Schwarzschild (or even Kerr) black holes. It is a dynamic situation. In practice, what is done (in numerical computations) is to define the surface of an apparent horizon, from within which, light rays appear not to be making their way outwards towards a distant observer.
The location of this surface (or surfaces when the black holes are well separated) must be calculated dynamically, and it changes as the merger progresses. After the merger it settles down to approximate the event horizon of an eternal Kerr black hole (a merger remnant will always have some spin).
However, the root of your question is the apparent paradox around the simpler situation of how anything can fall into a black hole if time dilation slows this process infinitely at any (apparent) event horizon. There is no need to try to resolve this paradox (it isn't a paradox, because there is no one "truth" of what happens in relativity, only what different observers observe) because the apparent horizon is dynamic (it moves) and objects that get close to the horizon become unobservable to a distant observer.
If you have two black holes, then the event horizons are distorted.
As event horizons are regions of spacetime (not just spheres in space) there is no issue with "time stopping". That is to say, while there is infinite time dilation for an observer at the event horizon, relative to an observer at great distance from the event horizon, the event horizons themselves are not "things". The shape of the event horizons is determined by the solutions of the General relativity equations, and the event horizons for two black holes are not spherical.
So the event horizons merge in space time, and the merged blackhole rapidly (over a few hundreds of milliseconds) settles down to a state that is asymptotically the same as a single rotating, "Kerr" black hole.
The key confusion here is thinking that event horizons are "things in spacetime" which are therefore subject to Relativity rules like time dilation. They are not. By way of (weak) analogy (and only to illustrate that relativistic rules only apply to "things") consider how the dot of a laser pointer can move faster than light.
In order to understand without analogy, you need to solve (numerically) the GR equations. Look at this picture of merging black holes. And remember, that the surfaces you see are merely 2d representations embedded in flat 3d, of a horizon that is actually embedded in curved 4d spacetime.
(image credit to Ligo:converted to gif by [user@uhoh])