Show, don't tell
While lambshaanxy's answer is a good one, it doesn't apply to all cases. It applies in cases where upper management is open to improvement.
But there are situations where upper management is either oblivious to or doesn't quite care about the workload of the staff (multiple levels down from them). I'm not implying malevolence here, just a matter of them being preoccupied with higher level decisions.
You've somewhat confirmed my suspicion when you said
I talked to my manager but he doesn't see the need for more team members.
Which seems like you're dealing with upper management that disagrees about the staffing issue and puts their opinion above that of someone who is closer to it (you).
You talked to your manager, and you focused on a concrete issue. However, it's possible that this manager has been actively managing the staffing themselves and therefore is unwilling to let you override the decision they already made. Again, that's not necessarily malevolent - though it can be construed as pride/ego depending on the situation.
If your manager is intent on making the staffing calls and ignores anyone else's input1, then they should also feel the repercussions from the decision they made (i.e. badly staffing the company).
1 I'm not saying that management is in the wrong by making their own decision. There are situations where they are actually right and you are wrong, where overriding you is the correct decision.
However, those who make the decision should experience the fallout from making a bad decision - otherwise how will they ever know they made a bad decision and learn from it?
You can do what you can do
You and your team member can only do so much in the hours that you're contracted for. If management piles more tasks on top of that upper limit, then that's just a pile of tasks that's not going to get done on time.
Report that the extra work can't get done within the expected deadline, ask which work takes priority, and start tackling the work in descending order of priority at a reasonable working pace.
As an aside, if "everything is high priority", then nothing is. If you cannot get an answer on priority, then it's up to you to tackle the work as you see fit.
When asked why the additional (lower priority) tasks were not done on time, show them a fully tracked worklog of what you have been working on, and how there weren't any more hours in the week to actually take on these extra tasks.
There are two possibilities here on how the manager responds.
"So what would it take to get it done?"
This is the part where it was never malevolent mismanagement.
They engage you in an actual discussion, and now have concrete proof that there is not enough manpower to handle the requested workload. An open discussion on how to fix the staffing issues can be had.
It's possible that you're only going to get one extra team member, which may mean that this cycle will repeat again, but if that's what it takes to get management on board, that's what it takes.
"You dropped the ball!"
This is the part where it was malevolent mismanagement.
They resort to blaming you and/or your team member for not getting the work done, not doing enough overtime, having caused the delays, or just ignore your feedback and just tell you to do better next time.
If they see the concrete proof that your working hours have been filled with tasks and yet deny this proof in favor of laying unfounded blame, then this company has a severely toxic attitude (at least in regards to how your team is treated).
My suggestion would be to look for another job. No amount of effort is likely to solve toxicity coming from higher up. You can try and be the umbrella that shields your team, but an umbrella doesn't stop the rain and it will eventually break down or wear out.
That's not to say it's impossible to turn things around, but it'd require you having to take on actual adversaries who outrank you, on top of you being a new hire. The odds of you coming out on top are astronomically low.