Can I use a balance remaining from my first 180-day period and add 90 days from my second 180-day period, without leaving the Schengen area?
It's a sliding window, not a period
You may be thinking the 180 days are Jan. 1-July 1, or 6-month anniversaries or something. Not at all.
The 180 days in question are starting from 180 days ago. So August 20 through today (Feb 15). Tomorrow, the 180 days will be August 21 til tomorrow (Feb 16).
If 90 of those 180 days have been in Schengen, then you cannot enter Schengen today. If 60 of those 180 days were in Schengen, then you can spend 30 more days in Schengen as of today, and this gets to be re-evaluated day by day as past Schengen days may scroll off.
So for insatnce if you spent 88 days in Schengen, then spent 100 days outside of Schengen, then 80 of your last 180 days were in Schengen. But you can still stay for 90 days, because for the next 80 days, an in-Schengen days will slide out of the window as fast as new in-Schengen days accrue. So your day-count will remain at 80 until near the end, when the old in-Schengen days have entirely slid out of frame, and then only the new ones will count.