Hi Don,
In addition to the other very valuable insights and suggestions you've received, for the specific problem of the carrier behaving as an "AM transmitter", I'd suggest you check the balanced modulator and/or the filter as the carrier without modulation could not get pass through these stages. On the other hand, check if there is no audio oscillation which could be getting into the modulator and them the transmitter would be legitimately send the tone as it were CW (in fact for working in CW some BitXs have a small audio oscillator which is fed in the modulator chain to that effect).
HTH
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Cesar Rabak, PY2CSH?