In this situation, would pitot-static instrumentation behave as if only the static port was blocked? What about if only the drain hole becomes blocked?
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A blocked static port will make the altimeter and vertical speed indicator inoperative. The airspeed indicator will function correctly at the altitude at which it first became blocked, but it will read high at lower altitudes and low at higher altitudes, since it's comparing the ram pressure to the wrong static pressure.
Because dynamic pressure is small compared to static pressure, this is likely to make the airspeed indicator essentially unusable if the altitude is changed too much. You can restore operation of all three instruments by using an alternate static port. If you have steam gauges, it is also possible to restore the altimeter and ASI by breaking the glass on the VSI.
If you can not unblock the static port, it is prudent to consider the airspeed indicator inoperative and cover it if possible. If you are on a stabilized approach with a blocked static port, the indicated airspeed will increase as you descend into denser air. The natural response to this will destabilize your approach and is asking for a stall.
The drain hole is meant to allow anything stuck in the pitot tube to escape. A blocked drain hole has no significant effect at first. Perhaps a small increase in indicated airspeed since a drain hole would tend to decrease ram pressure slightly. But if moisture and debris is allowed to build up it will eventually render the airspeed indicator unreliable or inoperative.