Is time slowing down and disappearing from the universe instead of the expansion of universe? Are we fooled into thinking that expansion of the universe is accelerating, when in fact, time itself is slowing?
Or if dark energy does exist?
 A: It is not so easy to give a definition of time. We see things change when time changes, time is the fundamental measure of the evolution, which could be a local evolution, or the evolution of the entire universe itself.
Say that time is "slowing" would mean that there is another time-like quantity (distinct of time) which  would be more fundamental,  but, by definition, if time is a fundamental measure of the evolution, this is not possible.
What you can do is the following : take an acceptable evolution variable, and use it instead of time. By acceptable evolution variable, I mean a variable $a(t)$, such as there is a bijection between $t$ and $a(t)$, for instance, such as $\large \frac{d a(t)}{dt} >0$ 
You could now express physic laws with $a$ instead of t, if you wish, but it does not mean that $a$ is the fundamental measure of the evolution.
For instance, in a expanding universe, the physical distance between 2 points increases with time, and in a class of models, the ratio of the physical distance at times $t_1$ and  $t_2>t_1$ depends only on time  and could be written : $\large \frac {\Delta x(t_2)}{\Delta x(t_1)} = \frac{a(t_2)}{a(t_1)} > 1$, where $a(t)$ increases with $t$. So here $a(t)$ is an acceptable evolution variable.
Note that the fact that the universe is expanding $\large \frac{d a(t)}{dt} >0$, is a different thing that saying that the expansion of the universe is accelerating, which is $\large \frac{d^2 a(t)}{dt^2} >0$
Dark energy (also seen as positive cosmological constant) is the origin of the acceleration of the expansion of the universe. (so dark energy is not at all contradictory with the acceleration of the expansion of the universe, as you suppose)
