Sounds like the lamp current is too high.
The wiring may depend on which country you are in; either 110 volt or 230 volt.
For 230 volt, there is probably one ballast choke and the two lamps are in series with each other and the ballast. A starter switch is connected across each tube through the lamp filaments. Not sure how they do it on 110 volt systems.
There may be lamp current passing through the filament due to a faulty lamp connection/connector. This would be wrong. In correct operation, no lamp current passes through the lamp filament, only at startup does current flow through the filaments and the starter switch.