Is it simply because the H instantly disassociates so it is a proton with zero shielding?
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Many $\ce{-OH}$ and $\ce{-NH2}$ protons do have characteristic shifts. However, their characteristic ranges tend to be much broader than those of $\ce{C-H}$ protons.
For example alcohol chemical shifts ($\delta$) range from roughly 1-5 ppm while pheonlics range from 4-8ppm and carboxylic acid protons are from 10-13 ppm or so. The ranges are 3-5 ppm in width, while similar ranges for $\ce{C-H}$ are 1-2ppm in width. Amines and amides are somewhat in between these two in terms of width of the ranges. So, saying they are not characteristic is an unhelpful description of the actual situation.
The next (implied) question is why would the ranges be broader for these groups?
Chemical shift is a measure of electron density near the proton being measured. The magnetic moment of electrons 'shields' the proton from the external magnet, changing the amount of energy require for the quantum mechanical transition of nucleus aligned with field versus opposed. Normally, we consider electron density to be a reflection of the bonding arrangements. However, for $\ce{-OH}$ and $\ce{-NH2}$ protons, hydrogen bonding is common. As such, often times nearby functional groups or NMR solvent are also contributing electron density, increasing the variability of the observe chemical shift from case to case.
Finally, in some cases the hydrogen bonding is dynamically changing slowly compared to the NMR time scale. In this cases you will observe very broad peaks. For example, water in $\ce{DMSO-d6}$ (a high-viscosity solvent) occurs around 3.3ppm. In very old bottles of the solvent (with lots of atmospherically absorbed water), I have observed this peak as much as 1.5 ppm wide baseline to baseline, always with a peak at 3.3ppm. However, in $\ce{CDCl3}$ water tends to have a pretty sharp peak at 1.5ppm (presumably because of limited water solubility).
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Labile protons, such as alcohols and amines have protons which exchange with other labile protons (typically water), and what is observed in the NMR spectrum is an average chemical shift for these species. The actual position will depend on the relative concentrations of the exchanging species, and the exchange rates.
If you consider two protons in very slow (or no) exchange, you will observe two independent peaks. Two protons that exchange so fast that you cannot tell whether a proton is attached to one molecule or the other will appear as a single averaged peak, with a chemical shift at a population-distributed average of those two exchanging peaks. For protons that exchange at an intermediate rate, peaks will become broadened, and have their chemical shift move towards the peak that they are exchanging with.
Hence, alcohols, amines, amides in many common solvents will have peaks that appear broad or have coalesced with the residual water peak. So, for example, the normal residual water peak in chloroform comes at 1.6. As the amount of labile proton signal increases in the sample, either from more water or alcohols etc, the shift of residual water slowly moves to higher chemical shift. Initially, the water and other labile peaks will appear as separate, but broadened peaks. As the concentration of these increase, the peaks broaden more and move together. At infinite concentration of labile peaks exchanging with water, the peak will appear at 4.7ppm.
If you exclude all water from your sample, and other pathways of exchange, the these peaks will have very characteristic chemical shifts, just as other protons do.
