It could be a number of things.
It definitely points to the carburetors. One thing is they could not be seated properly, allowing causing a lean condition, thus the spiking rpms.
Could be a float issue? Sticking (the flooding) or way incorrectly adjusted?
You can leave the petcock set to "ON" without an issue. it's vacuum operated, so should only pull gas when the bike is on. My fuel gauge shows 100% full at around 80%. If my gauge shows half I get gas, usually. It likes to drop quickly from half for me.
1986 Kawasaki Vulcan 750
NGK Iridium Plugs #7803/DPR7EIX-9
Duralast Gold ETX15 AGM Battery
Coastered & Shaved
Metzeler ME880 [110/90-19, 170/80-15]
Balance Dampers Replaced
Rebuilt Forks w/ Progressive Springs
VN750.com Grill Cover
Emgo 23-92411 Handlebars
MOSFET FH012AA R/R
1990 Kawasaki Vulcan 750
1998 Honda VFR800 FI
2014 Honda VFR800F
1989 Pontiac Firebird Formula 350