While not of much use now, your primary method for staving off mineral crashes is proper planning beforehand. But, as I said, that's not much use now.
Have you scouted for other potential jump points leading out of Sol? I'm guessing the primary reason the tutorial recommends setting a minimum number of jump points for Sol is to avoid the exact situation that you currently find yourself in.
Other than that, I'd say your best bet is to hope you get lucky with geosurvey teams on otherwise empty planets (Assuming you've already checked Venus). I was actually in a situation similar to yours early on where Venus had 20mil corundium @ 0.1, Titan had 600k @ 0.1, and I was lucky in that I had an asteroid with 50k @ 0.8, but very low amounts of other resources. A combination of uncovering several thousand tons of corundium in some ruins and discovering a world with 2.5mil @0.8 with high levels of other resources was what eventually broke the shortage. My situation may have been a bit worse, actually, as I was suffering from multiple shortages: duranium, mercassium, and corundium. I was pretty much manipulating the build allocation for my mines and labs on a weekly basis in order to make sure I didn't run out of their respective resources. Duranium wasn't as limited, as I could clearly keep building labs/mines, but shipyard expansion was pretty much out of the question and I was frequently forced to spend my discretionary build rate on financial centers simply because I didn't have the duranium for anything else.
I'd have to see the exact situation to really give more direct advice.
Not sure about the harvesters, but my design is engineless and relies on tankers to periodically ship fuel back to my primary colony in the system.