This thing here allows for joystick emulation:
https://github.com/rdoeffinger/xwa_ddraw_d3d11/releases/tag/v1.4.9
I never tried that function since I don't need it but it should work and allows to adjust sensibility.

EDIT

There is a possibility a version of it is already packaged with the GOG.com version, there should be a DDRAW.cfg file, open it with the notepad and see if the option for joystick emulation is there otherwise just download the file above and replace it.

Hi Det_Bullock, I'm sorry for the very late reply. Thanks for the help, but I've found a virtual joystick software called vJoy that emulates a joystick (I've chose it over your suggestion because vJoy it's not for Star Wars X-Wing/TIE games only, so I could use it for other games too). I've used a mapping software called Universal Control Remapper to map the virtual joystick to my mouse so I could play the game. It's working, but the ship keeps rotating even when I'm not touching the mouse... Do you have any idea what I could do to fix it?

"rotating"?
Try disabling the rudder in the game control options.

Thanks, that fixed it! Out of curiosity, what exactly does the rudder option do?

Activates the fourth axis for roll, in X-wing Alliance like most other spacesims it's used for roll (hence why with the rudder activated and no real thing attached to it your craft was spinning), in some other spacesims (like Elite) and in simulators based on real planes is usually used for yaw.
In real fighter jets the stick controls the ailerons while the rudder pedals control the vertical fin on the tail (the name escapes me) that is used for long range turns while the ailerons are used for sharp turns by rolling to the side and pulling the stick.
Most desktop joysticks to avoid the use of pedals have an additional axis used by twisting the stick, the so-called twist-rudder.

X-wing Alliance was the first game in the series to support a fourth axis, full throttle support was introduced only in X-wing vs Tie Fighter and the DOS installments of the series only supported the two basic joystick axis.

I see, thanks a lot for the help!

Although this is an old thread, the problem persists, so I have written a guide which I hope will be of use. You can find the latest version in the Guides to XWA on the appropriate Steam-Community Website (I am not able to post links here). Please see there for the latest version and further and more detailed information.

In short:

1. Install vJoy as virtual Joystick driver and FreePIE as bridge between vJoy and your input device.
2. Configure vJoy Device No. 1 with X/Y/Z/Rx/Ry/Rz axis and 2 Buttons
3. Run the following script in FreePIE (File->New, Paste, Save, Press F5):

########## FreePIE Script Start ##########
## !!!!! REPLACE ALL OCCURENCES of "____" (four consecutive underscores )
## with four consecutive spaces or
## a single tab before attempting to run !!!!!

#Initialization
if starting:
____rmbKey = Key.X #Key bound to right mouse botton
____#Higher values = faster centering
____centreX = 0.015
____centreY = 0.03
____#Sensitivities
____sensX = 50
____sensY = 50
____sensZ = 10
____viewSens = 50
____#Timings
____system.setThreadTiming(TimingTypes.HighresSystemTimer)
____system.threadExecutionInterval = 20
____#Initial values of axis / latches /etc.
____x = 0
____y = 0
____z = -vJoy[0].axisMax
____rx = 0
____ry = 0
____timeKeyPressed = 0
____freelook = False
____viewLatch = False
____moveLatch = False
____rmbLatch = False

#Freelook via Numpad mapped to mouse while middle mouse button pressed
if freelook:
____if timeKeyPressed <= 0:
________timeKeyPressed = viewSens
____
________#rx += mouse.deltaX * viewSens
________if mouse.deltaX > 0:
____________keyboard.setKey(Key.NumberPad6, True)
____________keyboard.setKey(Key.NumberPad6, False)
________if mouse.deltaX < 0:
____________keyboard.setKey(Key.NumberPad4, True)
____________keyboard.setKey(Key.NumberPad4, False)
____________
________#ry += mouse.deltaY * viewSens
________if mouse.deltaY > 0:
____________keyboard.setKey(Key.NumberPad2, True)
____________keyboard.setKey(Key.NumberPad2, False)
________if mouse.deltaY < 0:
____________keyboard.setKey(Key.NumberPad8, True)
____________keyboard.setKey(Key.NumberPad8, False)
____else:
________timeKeyPressed -= system.threadExecutionInterval
________if timeKeyPressed < 0:
____________timeKeyPressed = 0

else:
____# Mouse axis as Joystick via vJoy-Driver
____if mouse.deltaX:
________moveLatch = True
________x += mouse.deltaX * sensX
________if abs(x) > vJoy[0].axisMax:
____________x = vJoy[0].axisMax * x / abs(x)
____else:
________x /= (1.0 + centreX)
____
____if mouse.deltaY:
________moveLatch = True
________y += mouse.deltaY * sensY
________if abs(y) > vJoy[0].axisMax:
____________y = vJoy[0].axisMax * y / abs(y)
____else:
________y /= (1.0 + centreY)

# recenter
if (not freelook and viewLatch):
____rx = 0
____ry = 0
____viewLatch = False

#keyboard override mouse
#if (moveLatch and keyboard.getPressed(Key.Space) or keyboard.getPressed(Key.Z)):
#____x = 0
#____y = 0
#____moveLatch = False

#right mouse button
if (mouse.getPressed(1) or mouse.rightButton):
____if (not rmbLatch):
________keyboard.setKey(rmbKey, True)
________keyboard.setKey(rmbKey, False)
____rmbLatch = True
else:
____rmbLatch = False
____
#code for hold freelock
freelook = mouse.getButton(2) or mouse.middleButton
#if (mouse.getPressed(3)):
#____viewLatch = True
____
#code for toggle freelook
#toggle = keyboard.getPressed(Key.CapsLock)
#if (toggle):
#____freelook = not freelook

#Throttle via Z-Axis mapped to mouse wheel
if (mouse.wheel):
____z += mouse.wheel * sensZ
____if abs(z) > vJoy[0].axisMax:
________z = vJoy[0].axisMax * z / abs(z)
________
##### vJoy mapping #####

### Standard mapping ###
#vJoy[0].rx = rx
#vJoy[0].ry = ry
#vJoy[0].x = x
#vJoy[0].y = y
#vJoy[0].z = -z

### X-Wing Alliance default "weird" mapping ###
vJoy[0].rx = 0
vJoy[0].ry = 0
vJoy[0].rz = x
vJoy[0].x = 0
vJoy[0].y = -y
vJoy[0].z = -z

#Joystick buttons mapped to mouse buttons
vJoy[0].setButton(0, mouse.getButton(0) or mouse.leftButton)
vJoy[0].setButton(1, mouse.getButton(1) or mouse.rightButton)

#Diagnostics
diagnostics.watch(vJoy[0].x)
diagnostics.watch(vJoy[0].y)
diagnostics.watch(vJoy[0].z)
diagnostics.watch(vJoy[0].rx)
diagnostics.watch(vJoy[0].ry)
diagnostics.watch(vJoy[0].rz)
diagnostics.watch(freelook)
diagnostics.watch(moveLatch)
diagnostics.watch(mouse.getButton(0) or mouse.leftButton)
diagnostics.watch(mouse.getButton(1) or mouse.rightButton)
diagnostics.watch(mouse.getButton(2) or mouse.middleButton)
diagnostics.watch(vJoy[0].axisMax)

########## FreePIE Script End ##########

Once you have completed the above steps, you should be able to control your ship with your mouse movements, use the mouse wheel as throttle, fire your weapons with the left mouse button, (un)link your weapons using the right mouse button and free-look while holding the middle mouse button.

What I like about this solution is that it is not limited to a specific game or even gaming. With FreePIE you can work with the RAW inputs and have full control over all operations. Of course, you can change the example script to make the inputs softer by setting centerX and/or centerY to lower values - or discard automatic centering altogether.

According to the FreePIE documentation, you can also use this method to modify the inputs of your hardware joystick, transfer them to vJoy and thus solve problems such as wrong axes or other inconveniences.

What can be inconvenient with this solution is that you need some programming knowledge to make customizations.