How do I cook a steak well-done without it drying out?
sous vide is an excellent technique to use, for your situation.  You can cook any number of foods to your desired doneness while also pasteurizing the product.  So, you can have your cake and eat it too, so to speak.  With sous vide there is no problem with a rare steak, because you can pasteurize.  Remember, pasteurization is not simply a result of temperature...but, temperature + time.  For example, with sous vide, you can cook a raw egg at 57C (134.6 F) for two hours...it will appear raw and behave as raw, but will be pasteurized and safe to eat.  Why give up steak the way you like it?