This is the basic problem: the minute that there is a predictable way to avoid punishment, the deterrent effect of the punishment disappears. So there is not, and should not be, an obvious, easy and predictable way to avoid punishment.
You have two obvious choices: to 'fess up and hope for a lesser punishment, or to wait and see if they actually catch you. From the perspective of those of us who administer punishment to students who cheat, we are always inclined to be less severe with those who confess. Part of this is that by encouraging confessions, we hope to not create an environment where it's always to students' advantage to wait and see if they are caught. But if, again, it becomes predicable that we will be very lenient with those that confess, then what we encourage is cheating followed by confession with increased expectations of leniency. And maximum leniency is no disciplinary action, and that just brings us back to the basic problem of deterrence. The comparison group is not those that did not confess, but those who did not cheat. Are we being fair to those who did the work themselves? That's the question.
You mention that you are really sorry, and that you just want to learn from your lesson. The problem is that you are likely lying to yourself, not on purpose, but still. Let me assure you that 100% of the time I am lenient to a cheating student, they do it again, only that this time they assure me that they are not 100% sure, but 1,000% sure they'll never to it again. We don't want to punish students, we want them to stop cheating. Thus, the only way to reach our true goal is to administer a punishment that "hurts." Hurts in the sense that it creates enough discomfort and regret, that it prevents the act form happening again.
If you really want to stop yourself from doing this again, then confess and accept whatever disciplinary action they take against you. It has the added benefit that it might bring, albeit not guarantee, leniency.
Last, try to keep a balance between feeling bad for what you did, and overdoing the self-recrimination. Cheating is academia is rampant and widespread. Before LMLs came about, it was Chegg, and before Chegg, it was paying Internet "tutors" to do the work for you, and so on. Only the technology has made it easier to cheat, but I really doubt that students in the 80s where less inclined to cheat: they just didn't have the technology, or were never caught (look at how many theses from famous people are being found to be plagiarized now that automatic tools are available.) There's plenty of guilt to go around: from professors who never check for plagiarism or just test for things that can be answered with a quick Google search, to ever-lenient administrators never wanting to "punish" a paying customer, to a general devaluation from students about what constitutes an education. We are trying to train you here, so please, do the training exercises. Like I ask my students caught cheating: Do you also pay someone to go to the gym for you?
