My wife and I are fast approaching our fifth "anniversary" of taking ballroom lessons, so I do have some understanding of your situation. Here is our "method" of dancing:
We started off just taking private lessons together, as a couple. After a few months, we both decided that we would like to try competitions. Also, we decided that we wanted to dance competitively in pro/am respectively, plus as a amateur couple. This decision meant that we would have to each take separate private lessons with our respective teacher, and also take lessons together.
Our method continues to this day. We both take lessons separately with our respective teachers and together with our "lead" teacher. In our case, my wife has far exceeded my ability, due partially to the difference in ages (I'm in my 60's, and she is quite younger) and work schedules (she has more time). We have completed 12 competitions and we are packing to go to another competition next week.
My wife is definitely better than I am, having gained "silver" level in many of her dances, competing in both International Ballroom and Latin plus American Smooth and Rhythm. In fact, she just returned from competing in the NDC (National Dance Championships) last week. Meanwhile, I just "struggle" along with American Smooth and Rhythm. She takes about 3 to 5 private lessons a week plus group dance classes and I do 2 privates a week plus group classes, plus 2 private lessons a week as a couple. With her experience, she has taught me how to lead better because she understands how to tell me what my body movements signal to a lady.
Through it all, we have managed to maintain our excitement of dancing together, still competing as a couple, and being awarded Best Amateur Couple in many of our competitions. Yes, we both have very competitive spirits, but we can also channel our competitiveness separately or together.