Any soil with sphagmum peat moss as the major component REQUIRES lime to buffer the soil to the correct PH (ie. PH of around 6.0). Peat moss is organic and constantly breaks down releasing acids in the process, so it tends to have a rather low PH by itself (PH of around 4.0). So if you used a peat based soil mix without lime, you'd be growing in a medium with a PH of 4.0, which would provide very poor growth for most all plants with the exception of acid-loving plants (not cannabis). This is why almost all commercial soil mixes contain calcitic and/or dolomitic limestone (the former provides calcium, while the latter provides both calcium and magnesium). Some people add more of their own lime to the soil in addition to what already in the soil from the manufacturer, this is to provide a buffer over a longer period of time, but too much will toxify the soil (like too much of anything).
I'm afraid your problems have nothing to do with the lime in the soil nor the PH of your water. Water PH has very little effect in buffered soil, that's the whole point of a buffer, whatever the PH of the water you're adding, be it 5.6 or 8.2, it will be buffered by the soil mix to around 6.0. Of course, it doesn't hurt to PH your water to around 6.3 just to keep things in balance. Maybe you could post a picture to make your problem more clear.