Is there a tactful way for me to tackle this issue directly?
There's a lot going on here. First off, I wouldn't rule out any issues such as dyslexia that may make writing 'correctly' hard for your group mate. And on your side, from your description of yourself there's a real chance your social awkwardness didn't help.
One of the ways I found you can give feedback in both school and professional life is by asking questions and opening up conversations. There's a world of difference between telling someone 'your writing sucks, please take some care to improve it' and
We noticed you seem to struggle with writing good/correct English, and when we notice that we worry that your actual work won't be valued as much, which would be a shame because it's good! Is there anything we can do to help improve your writing and make it easier for teachers to recognize the quality of our work?
An example where this worked: I've had coworkers that would often break 'our part' of the build, just because 'their part' was still fine. Instead of giving feedback that said 'you need to test better', we opened up a conversation about what we could do to help. One of the things we did was untangle some code, the other was that we made a list of things he had to keep in mind when testing because we simply couldn't untangle them.
Sometimes, there's a lot of 'fluff' needed for giving feedback. Othertimes, you can be more direct. What approach you use really depends on both the person giving feedback and the person receiving it, and the interpersonal relationship between the two. Some people prefer giving or receiving feedback sandwiches. Others might prefer receiving more direct feedback because the compliments feel fake, but they still e.g. appreciate the other person using I-messages so the feedback doesn't sound accusatory.
In your specific case, I would recommend giving the feedback in such a way that it allows for your group member to just ask for any help they might need to overcome their writing issues and increase the quality of your team's work. Be open to offering at least some amount of help. I understand you don't want to/don't have the time to rewrite everything, but if you can help by installing a spell-checker for them, that's a great outcome!
Given that you're dealing with someone who's reacted negatively to you and your feedback before, perhaps you can get your other team partner to lead the conversation. After all, they seem to have just as much trouble with this writing problem as you.
This might be a better option before asking a professor to fix it for you: my teachers would usually be very willing to help, but only once a group made an effort to sort things out themselves and it really didn't work. For example, I had a history teacher that only interfered and split our group once we showed two of the group members weren't doing anything, by requesting them to bring some books from the library in front of said teacher. When the books weren't there and we were just sitting in class doing nothing because we had no books, we were finally allowed to split the group.
The most difficult part here will be deciding on the who, where, and when. Two people giving feedback to one person can be intimidating. One person giving feedback on behalf of a group may face disbelief. Stopping someone in the middle of a hallway or taking them apart from a group/room may make people more nervous than just casually working the feedback into a conversation about the group project that's already going on.
I usually prefer dealing with single people and picking a more casual moment because that's how I like being dealt with, but I try to take note of clues like people reacting in a hushed voice to my feedback or that start fidgeting: At that point, it may be a sign they're uncomfortable because the place is too public for them. I think here the communications professor may be of help, especially if this professor also works with your other teammates they may have picked up on some clues that can help in making this decision about who, where and when.