How to deal with the units of quantities if $\hbar=\tfrac{h}{2\pi}=1$?
For example, the energy $E=\hbar\omega$:
If I have chosen $\hbar=1$, how do I use the units to properly differentiate between energy and frequency, or any other physical quantities?
If you're setting $\hbar=1$, then you don't - you can't - distinguish between energy and angular frequency. They are, in fact, the same quantity, since $E=\hbar\omega=\omega$. Similarly, if $\hbar=1$ you can no longer draw dimensional distinctions between wavevectors and linear momenta, or between angular momenta and pure numbers.
In general, you only set $\hbar=1$ if this is a desirable state of affairs for what you're doing, and the convenience in dropping constants is large enough to offset the loss of one dimension you could otherwise use to perform dimensional analysis.
In the case of energy, I find it more useful to use the reciprocal of time rather than frequency. Time is a much more fundamental (and common) dimension. In other words $E=1/T$, and of course, $T=1/E$.
