The way that moving around changes the pitch of the sound indicates that there is some bug in WebGL doppler effect simulation. This is possible, as WebGL uses different code for audio then other platforms. I’m not aware of any current issues in WebGL doppler effect simulation (but there have been some in the past). If you file a bug report with a project folder (and quote the case number here), I’m sure we can find a fix quickly.
FYI, we found the source of this issue (Doppler effect simulation is not correctly disabled when setting “spatial blend” to 0 on WebGL). This will be fixed in 5.3, until then, the workaround described by @AnonDreamer is fine.