Variable band-pass filter for splitting audio signal into separate signals (high and low freqs)
is there any way of creating a continuous (not sure if it's the right
word, as in not discrete, allowing for smooth variable change instead
of choosing from a constant number of options) variable low-pass and
high-pass filter?

It sounds like you are looking for a parametric equalizer: -

Parametric equalizers are multi-band variable equalizers which allow
users to control the three primary parameters: amplitude, center
frequency and bandwidth. The amplitude of each band can be controlled,
and the center frequency can be shifted, and bandwidth (which is
inversely related to &quot;Q&quot;) can be widened or narrowed. Parametric
equalizers are capable of making much more precise adjustments to
sound than other equalizers, and are commonly used in sound recording
and live sound reinforcement. Parametric equalizers are also sold as
standalone outboard gear units.
A variant of the parametric equalizer is the semi-parametric
equalizer, also known as a sweepable filter. It allows users to
control the amplitude and frequency, but uses a pre-set bandwidth of
the center frequency. In some cases, semi-parametric equalizers allow
the user to select between a wide and a narrow preset bandwidth.

They are commonly available and, if you don't want to build one you can buy one like this stereo model: -

Feed both channels with the same common input signal and adjust each parametric equalizer to give you the response you want. You can find tons of circuits on the internet for them too: -

Schematic from here.