Your guess is basicly right.
On the backside of the phosphor in a CRT there is a layer of aluminium, which reflects the
light emitted backward to the front, blocks positive ions from damaging the phosphor
and is used to carry of the electrons after they have done their duty. This aluminium
layer is connected to the anode ("plate") terminal.
For this reason the beam is not the cause of the electric charge on the front suface
(which You describe as "fuzzy feeling")
This charge is produced by influence ("electrostatic induction").
The inner wall (that aluminium) is kept at some 10 to 30 kV positive during
operation. The charge induced from that layer to the outside stays there if t
here is no special surface treatment. Slowly it can creep away, but when You switch off,
a induction of opposite polarity arises. (this is often accompanied by a
rather loud "hiss" (corona).
On top of that the glass front can leak small currents from the anode voltage,
this complicates analysis further.
Because the surface charge of CRTs can "load" dirt particles with charge and those fly then away with some "push", the CRTs of PC monitors were treated to be conductive starting
from about mid-nineties. (A lot of people complained about problems with burning/tears in eyes etc.) For some time one could buy frames with a fine metal mesh to be positioned in front of monitors of the first generation.