The answer depends on the circumstances: how do you change the resistance? Both the drift velocity and the number of available charge carriers can be changed.
In a basic Drude model for electrical transport both, $n$, the charge carrier density and $\tau$, the time between collisions determine the resistance:
$$\mathbf{J} = \left( \frac{n q^2 \tau}{m} \right) \mathbf{E}.$$
Here, $\mathbf{J}$ is the current density and $\mathbf{E}$ the applied electrical field. The term in parentheses characterizes the material properties, i.e. its resistance.
By changing to a different material you can influence both, $n$ and $\tau$, so both cases are possible.
The answer to your question is "yes".
The relevant relationship is $I= nqAv_d$ where $I$ is the current, $q$ is the charge on a single charge carrier $n$ is the number of charge carriers per unit volume and $v_d$ is the magnitude of drift velocity.
In a series circuit the current is constant as a consequence of the law of conservation of charge.
This means that the amount of charge per second entering a conductor at one end must equal the amount of charge per second leaving the conductor at the other end.
Charge is not created or destroyed within the conductor.
To simplify matter let’s look at the variables one at a time and assume that the current is the same in a complete circuit or part of a circuit.
Keeping the area and the charge on a charge carrier the same means that if a conductor has fewer charge carriers per unit volume the drift speed must be larger than the drift speed in a conductor with more charge carrier per unit volume.
So how is this increased drift speed achieved?
It is achieved by having a larger voltage (per unit length of the conductor) across the conductor which has the fewer charge carriers per unit volume.
So if you had a piece of copper and piece of iron (a worse electrical conductor that copper) which had the same dimensions as the copper in series and you passed a current though them the voltage across the iron would be larger than the voltage across the copper.
Put another way the resistance of the sample of iron is larger than that of copper.
This is your resistor and copper wire situation.
Suppose now there were two copper wires connected in series of the same length but one piece of wire had twice the cross sectional area of the other.
To transport the same amount of charge per second through both wires the drift velocity in the thinner wire must be twice that of the thicker wire.
You could say that there were in total fewer charge carrier in the thinner wire than in the thicker wire and so to convey the same charge per second the charge carriers would have to move twice as fast.
How is this extra speed achieved, again by having a larger voltage across the thinner wire that the thicker wire.
The resistance of the thinner wire is greater than that of the thicker wire.
You can do a similar analysis comparing the charge on the charge carriers if they are not the same.
All this you can convert to the analogy of a flow of water with the pressure difference being the analogy of voltage and the length and cross-sectional area of pipes being analogous to the length and the cross-sectional area of the conductor.
Usually it is the volume flow per second which is cited as the analogue of current but that means that there is a problem with finding an equivalent to the number of charge carries per unit volume in the electrical case.
So it is better to say that the mass transported per second is the analogue of current.
How this is done in practice I do not know. Perhaps it has to be shown as two separate circuits?
Anyway is you have two pipes of the same dimensions and need to transport the same amount of mass per second through the pipes then the liquid with the smaller density (number of charge carriers per unit volume) would have to travel faster (drift velocity) and so there has to be larger pressure difference (voltage) across the pipe with the lower density liquid flowing through it.
An finally as a caveat to my initial answer to your question it is theoretically possible by juggling the area of the specimens (resistor and wire) to make the charge carriers in a resistor move slower than that in a wire but in practice this is not possible because the charge carrier density in copper is so much greater than that of the substances used in resistors.
Best Answer
If the current is larger more charge flows through the resistor per second.
What you are missing is the fact that the current through your resistor is controlled not only by the resistance of your resistor but the circuit elements of a complete electrical circuit of which your resistor is one part.
Your resistor not only has an effect on the current passing through it but also the currents passing through other circuit elements to which it is connected.
In the end as there is no source or sink of charge within your resistor, the current entering your resistor coming from the circuit to which it is connected to is equal to the current leaving the resistor and that current is going into the circuit to which your resistor is connected to.