Simply centre punching and then drilling is what caused your problem. After centre punching, and with the tang and the trigger plate out of the stock, use a drill press to pre-drill those holes well undersized. Now you have a guide that your tap drill is going to want to follow, when you drill between centres. As Rich has wisely advised, when you are between centres, drill from each side only half way with the tap drill. Remove the trigger plate and drill down through the tang with the clearance drill bit.
In your botch, you have only said it is not centred well, but a picture would better describe how far off you are. If it is only slight, I likely wouldn't bother, chalking this one up to experience, but if it's way off centre, then a patch and re-drill is in order. Drilling a second hole either forward or to the rear of the old one is not the answer. Your tang screw must pass through the tang at 90 degrees, otherwise, it will be cocked crooked and look aweful. Drilling new holes fore or aft will make this happen for sure. There is only one place for that screw to pass through the trigger plate, and that is ninety degrees to the tang.