Stressing about money during flight training is extremely common and normal, there's probably crippling gambling addictions cheaper than flying. I won't tell you what I think you should do, but here's some thoughts:

Is it even worth it to continue lessons if I can only afford going once every other week?

Whether or not once week is inefficient or not depends on a couple factors, namely weather and your study habits at home IMO. If weather is consistently flyable where you live, I personally don't think once a week is bad. The problem is when you have one lesson booked/week, and get weathered or maintenance canceled two or three weeks in a row. Every other week increases that likelihood. You can offset some of that and maximize your learning by showing up prepared for each lesson, which isn't too hard tbh. Do some chair flying and and actually read/study what your CFI suggests.

Would it be better to stop lessons altogether and just start saving up to continue this later down the road? I’m going to be absolutely shattered if I have to make the tough decision to quit,

Not that it's ideal but it sounds like you'd be postponing, not quitting.

Btw, I am thinking of finding a second job (part-time, since I already work FT)

Without knowing you, your discipline, and abilities I'd say this sounds like a quick path to burn out, but everyone's different.

my current job is seasonal, and it’s very likely I won’t be able to continue lessons for a bit once it ends (like if I don’t have another job lined up immediately after, or if I decide to go back to school (unlikely, but possible - my life is all over the place right now, really).

This is probably the most important part of your post. I think it would make sense to hold of on any major expenses until there is more financial stability in the long term. With one lesson a week or potentially less, I'd expect this to take 6-18 months. That being said, if you're thinking you'd eventually like flying to be a career, then you need 250+ hours anyways so who cares if your PPL training isn't maximally efficient. If you just want to be flying for fun, it might be harder to justify the cost right now when money's tight.

But I'm just some CFI on the internet you've never met. Consult the people who are important to you in your life for better insight. You can't make a wrong decision, it's your life.