try changing dosbox's joystick options

cycles=20000, timed=true

Please forgive my stupidity.

How does one change Dosbox's joystick options?

read the manual

The only thing mentioned in the manual is how to use the keyboard to simulate a joystick (Don't want to do) and how to switch around the axis (again, don't want to do). There is nothing mentioned in the manual about correcting for drift.

you change Dosbox's joystick options in Dosbox's options...

How does one change Dosbox's joystick options?

Dosbox is configured by editing the <drumroll> configuration file. The manual (README.TXT) tells you how to edit and where to find the configuration file:

The file is divided into several sections (the names have [] around it). Some sections have options you can set. # and % indica […]
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The file is divided into several sections (the names have [] around it).
Some sections have options you can set.
# and % indicate comment-lines.
The DOSBox configuration file contains the current settings.

DOSBox will parse configuration files that are specified with -conf. If none were specified it will try to load "dosbox.conf" f […]
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DOSBox will parse configuration files that are specified with -conf. If
none were specified it will try to load "dosbox.conf" from the local
directory. If there is none, DOSBox will load the user configuration
file. This file will be created if it doesn't exist. The file can be
found in ~/.dosbox (Linux) or "~/Library/Preferences" (MAC OS X).
Windows users should use the shortcuts in the startmenu to find it.

The config file has a section called [Joystick]. You can change joystick options there. The option you want to change is either joysticktype or timed. Changing timed to the other value (can't remember whether true or false is the default) will probably fix things. If it doesn't, change joysticktype to something else (probably 2axis - the drift might be caused by the extra axes).

There is nothing mentioned in the manual about correcting for drift.

Actually, there's this:

Q: The cursor always moves into one direction! A: See if it still happens if you disable the joystick emulation, set joystick […]
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Q: The cursor always moves into one direction!
A: See if it still happens if you disable the joystick emulation,
set joysticktype=none in the [joystick] section of your DOSBox
configuration file. Maybe also try unplugging any joystick/gamepad.
If you want to use the joystick in the game, try setting timed=false
and be sure to calibrate the joystick (both in your OS as well as
in the game or the game's setup program).