Follow TCAS for vertical, and ATC for headings
Modify that.
Never turn when TCAS is issuing instructions. The TCAS software solution is based on 'wings level' and commanding rate of climb or descent.
Turn during an RA or TA and you throw your protection out the window.
"Traffic" or RA? Follow instructions. Don't turn. RA trumps ATC. Advise ATC iaw local AIP.