How do I resample an audio signal without altering its pitch?
Simple resampling would not work in your case. Resampling, while stretches/compresses the signal in the time domain, also does an inverse operation in the frequency domain.

Some basic time stretching algorithms you might want to look into (which do not result in pitch change) are:

1) OLA - overlap and add

2) PSOLA - pitch synchronous overlap and add

These work well for monophonic signals if that's what you have.

3) Phase vocoder

This works for polyphonic signals as well. But you may have to be careful about transients while using a phase vocoder approach.

There may be time stretcher plugins you could look at which use these. You could also find some open source code and time shift the signals in MATLAB/Python, etc.

Hope this helps. Let us know how it works out!