There's a few important things to consider to address your question
1. Does the dual degree MD/MPH program at your school require an additional year of schooling beyond the 4 years it takes to complete the MD? If the answer is yes, then read on. If the answer is no, then I would apply as a dual degree student
2. Is the university you will be attending for medical school have a MPH program of their own separately? If so, it might be worth calling their admissions office to see if any of the medical school credits. If the answer is yes, then starting this MPH program once you are in medical school is a better idea. Many of these programs can be completely online and started/stopped at your convenience if it is at the same institution you are completing your medical degree at
3. If the university does not have its own independent MPH program....this is where I would probably take the extra year and apply to the dual degree program. If you apply to another university for the MPH that doesn't accept any of your MD credits toward the degree, I feel as though you will be spending a lot of money that you don't need to spend
To address the competitive question, at my school, there are not a select number of MD/MPH spots so it does not affect competitiveness. For a program like MD/PhD on the other hand, there are only a few allotted spots making it more competitive. Again, I'm sure it differs for different institutions
Every program is different. Theres no harm in calling programs and asking questions!
Hope this helps!