This is horrible, really my worst fear, so my heart sank reading all of this. I'm very sorry that you're going through this right now. It's not an ideal situation to be in and there is no easy solution for it.
Someone can correct me if I'm wrong, and this may not even apply to you, but if you're living in the US and in a state where abortion is illegal, you're pretty much locking in your decision to carry by visiting a doctor. I'm also unsure of how long you even have the option of doing so in any country, I think maybe you may know better than me, but I also thought all of this was worth mentioning.
It is your boyfriend's kid too, so he should have a say and have his stance on this issue considered. But it's not just his baby, and it actually isn't his body at all. He got to do the fun part, but you have to do the heavy lifting even after the birth. The reality is that no matter how much a partner can and wants to help, most of the responsibility will fall on the parent who was pregnant. And on top of existing mental health issues, it can quickly become a disaster. You have a right to be worried and your desires and health should also be considered. If your bf is well off that's great, but it doesn't take just money to raise a child. A hurting and disinterested parent (especially mother) can be more detrimental to a child's development than poverty is, in my opinion. And leaving the baby here via ctb, while that will always be your choice and right, will be downright traumatizing. Not just for baby, but for boyfriend as well. You have no way of knowing how he may internalize your suicide after giving birth, and no way to tell or control how he will treat baby thereafter.
If he thinks that you are happy and excited, you two need to have a conversation about how you're really feeling. If you do decide to go through with having the baby, I think the general consensus will always be that you should stick around and make the child who did not ask to be here a priority instead of handicapping them severely with your intentional death. I'm sorry if that sounds insensitive to you as a parent and person, but it's how things are. Children are defenseless and innocent, they deserve the best chance at this life. With that being said, I do feel for your mental state and I have no doubts that things will get harder moving forward. Your boyfriend needs to be on the same page so you two can either decide that this is not the right time, or so he can be prepared to support you and baby in any way possible. A lot of things can happen, he needs to be emotionally and financially able to handle it. He will likely need a support system of his own and so will you, for either decision. But neither of them are wrong or right, you are simply choosing the one that is best for you.
If you do decide to go through with this pregnancy then yes, going to the doctor to get checked up is the next best step. You can also ask them for resources or referrals on how to deal with any depression or postpartum depression moving forward. I really hope that things work out for you, whether you decide to have the baby or not. I do strongly suggest you being completely honest with your boyfriend and seriously considering your options, though, as you don't only owe it to him and the baby but mostly to yourself.