Five months is a good amount of time to date. You know you like each other, or you probably wouldn't still be dating.
But you both have conflicting and busy schedules, and that makes it challenging for a relationship - or at least one in which you haven't set clear boundaries, both for your space together, or for the space you need for yourselves.
This sounds like a generally -generally!- simple issue to me. You don't have enough time for each other and yourselves, you want more for both, and you're dealing with the guilt trips and stress to and from each other as a result. This part is normal. And you *can* work around it, but you both need to be fairly lenient.
What should be a concern is how you communicate that. When you both have things to do for yourselves, as you said, it shouldn't be a point of contention, right? But it is. We humans tend to take our own feelings of inadequacy out on each other, so who knows, maybe he's punishing you for not feeling available enough. While the base reason is important, the ability to talk about it is paramount, so find out what's REALLY bothering him, make sure you address and avoid crappy texts that indicate dismissiveness like "Okkkkkkkkkkkk" - and then STOP trying to have difficult discussions like that in text to begin with. The minute it starts getting ugly in text (and we all read things different ways, though l agree with your interpretation of the text in question, and that you should have been at least talking on the phone to begin with), STOP. Say, "Call me, and let's talk about this in person", or "Let's calm down a bit, and then talk", or some other variation. There are statistics being collected on how many breakups occur through online or text interactions, and it's pathetic. Don't be a fucking statistic, and l mean that in the very kindest way.
Don't apologize if it's truly not on your part, but be very, very vigilant about examining your own part in it, no matter how small it may be; as you said, name-calling doesn't help. These things are important, and may be a fulcrum in how well your relationship moves. Remember you're in this for a reason, and if you don't *both* benefit from that reminder, then you're facing a problem.
And if his priority is having fun on top of all the priorities you both have to deal with (which is understandable, but obviously limited), consider whether your final priorities match up. lf you want a night in because you're always tired and he wants a night at the bar, these things should be considered.
But no, to answer your original question, you aren't losing it. You're dealing with real-life issues, real-life jobs, and real-life situations that occasionally need an outlet. This is true for all of us. Hang in there.