The best thing to do, in my opinion, is to just go with it & wait for things to settle down on their own. Depo Provera is a big dose of hormone for your body to get used to, as it's gradually released from the pool that the injection creates in your muscle. Your body is currently trying to get used to that, & in the meantime, you're having these hormonal symptoms.
If you start taking extra hormones in the form of birth control pills, then that's just giving your body more chemicals to deal with. It might work in your favour, or it might not. It may even stop the bleeding, but you risk even more side effects further down the line, for example.
Unless the bleeding is extremely heavy & constant, then I'd try to ride it out & give your body a chance to adjust. Most of the symptoms from going onto Depo Provera tend to subside once you've had 2 or 3 shots.
You don't need condoms with Depo Provera. It's extremely effective. Even more so than the regular birth control pills.
The Manufacturer advises that you shouldn't be on Depo Provera long term. There are risks associated with it, such as Brittle Bone Disease (Osteoporosis) & other risks.
If no other birth control suits you, then you definitely need to be taking a break every couple of years, & using a good calcium, magnesium & vitamin D supplement, whilst using Depo. One that gives the correct ratios of those 3 things, as that can affect how well you absorb the calcium into your bones.
If you put the name of this drug into the search field of this website, you can see the various risks associated with it.
If you can't stand the bleeding, or it is too heavy & constant, then your doctor can try more hormones, but otherwise I'd give them a miss, as it's just more hormones that come with more risks.