fix: Race condition when playing/pausing audio

[Gustl22]

Description

Fixes #1687

Instead we could also set the state always at the beginning of each method, but this raises these problems:

• If the operation in the method throws an error, the state doesn't match the native players state
• The state doesn't reflect the current real world audio state, as it's maybe still preparing/buffering and not actually playing

Checklist

• The title of my PR starts with a Conventional Commit prefix (fix:, feat:, refactor:,
  docs:, chore:, test:, ci: etc).
• I have read the Contributor Guide and followed the process outlined for submitting PRs.
• I have updated/added tests for ALL new/updated/fixed functionality.
• I have updated/added relevant documentation and added dartdoc comments with ///, where necessary.
• I have updated/added relevant examples in example.

Breaking Change

• Yes, this is a breaking change.
• No, this is not a breaking change.

Related Issues

#1687

[spydon]

Why do the methods need to be split up into a private and a public method? 🤔