Hi - lurker here but I made an account to respond to this. I would first ask why exactly you want to go into math academia to begin with? Judging by your coursework, I assume you were a CS major before you transferred. Going from that to an academia-focused math major is quite the leap.
My immediate practical advice would probably be to retake the courses you took at your old institution that you cheated on and take them honestly, so you actually have a chance to understand what's going on. Those courses are comparatively simple and straightforward to the upper-level classes in a typical math major, especially the harder versions that you would presumably take in an academia-focused track. If you struggled with those courses, you might forgive me for being unsure about how you will do in those harder courses. You might have to do that anyway given the grades you got.
This is also the part where I mention the IMO far more important part of all this, which is to speak from your ethical perspective. Firstly, it seems like you understand the moral implications of what you've done pretty well, so I won't waste time trying to berate you. I wouldn't dwell on it excessively, although I must also comment that I'm not sure what the point of your final note is. What does it matter if you were a member of a "long line of plagiarism"? It seems like a weird semantic point to make, and IMO a pointless one. Can you really say that you will never cheat again? You may be (presumably) on-campus this coming fall and remote learning and the pandemic might be (presumably) winding down, but if you are pressed with additional challenges, whether they be upper-level courses or the challenges of moving to a completely new institution, can you say for certain that you will remain untempted to cheat? After all, like you mentioned, the internet is right there!
I sympathize with the challenges you faced, and thus I think the much more important part of this is to talk to someone else you trust about all this, or even better a therapist. I can't speak for your academia prospects as I'm just an undergrad myself, but what I can
speak to is that school is far easier if you're in an adequate mental state to do schoolwork. I would not talk to your old professors about your cheating; rather, what it sounds like you need is a support system to give you what you need to succeed. The guilt you feel and the acts you've done don't have to define you, and you can certainly bounce back. I'm not giving you a free pass to say that what you did was morally alright, but what is more important now is how you proceed. So find and surround yourself with people you trust and who can make sure you do the right things. My DMs are always open if you need to talk, and certainly if you want my Discord you can DM and ask for that instead.
TL;DR: Reevaluate your coursework and yourself, get some help, and best of luck.