I haven't done a water change on the bigger tank for several months.
This is likely the reason your fish are dying. They have adapted to high waste levels and when fresh water is added they experience shock. By not doing water changes for months and months you allow a lot of dissolved organic wastes (from fish food/poop/decaying plants/dead bacteria) to build up in the water. Proteins, fats, and sugars do not fully degrade to nitrate in aquariums like they do in nature without long periods of time and huge bacterial loads.
These waste products cause long term stress in fish (lowering immune systems, stunting growth, damaging organs) which make them much more vulnerable to minute environmental changes. The wastes also promote algae growth (certain algae species directly consume organic wastes and use them to grow). This is the reason regular water changes are an absolute must. Not doing water changes to save effort does not work, a water change free tank is simply not possible at the hobbyist level.
So maybe in someone's book it doesn't matter that the pH is 6.0 or 7.6 but obvious my fish haven't read that book. Maybe it is the hardness. But I've also been told that shouldn't matter.
The pH will tend to rise as minerals from top-off water keep getting added to the tank. Evaporation loses water only allowing minerals to accumulate in the tank. These minerals and other dissolved compounds raise the pH over time. Also, if there is no CO2 system on the tank a heavily planted tank will often start to consume the carbonates in the water as a source of CO2 (the KH), this removes buffering capacity and further raises the pH.
For healthy fish pH fluctuations and other environmental changes are not a problem, but for a highly stressed fish it can easily be the straw that breaks the camel's back.
Why would I suddenly have no nitrates in my tanks but the ammonia and nitrites are absent? I've always had 5 ppm nitrates in my 30 gal tank. Suddenly, start topping off with RO water and they disappear, the fish all go into hiding and all my shrimp die off. But the plants are now fine and the algae is gone.
Plants use nitrates, 5 ppm is very low and can easily be used up in a day or two. This is why most people maintain a minimum of about 15 ppm nitrates for plant growth.
How did the water become hard in my 30 gal tank if it comes out of my tap soft?
Evaporation, followed by constant topping off without a proper water change.
I plan to just switch it over to RO water but obviously I haven't gotten the RO situation under control either seeing as how my betta is not doing so well in the straight RO tank. They have been in that tank for more than a month.
Using RO water may help your current situation, but what you need to do is start doing several small water changes a day (5-10%) of your tank's volume. These small water changes will remove small amounts of waste that have built up in the tank water and allow the fish time to adapt. After you have done this for at least a week, you can start doing regular 50% water changes once a week or once every other week.
Any photos of your tank?