Is it possible to just tear off the 1 portion that needs to be ripped off and just shingle over the other part of the house?
Yes as long as any local authorities who may have jurisdiction are OK with it.
Its a bit unconventional but it will save some money.
Trying to be cost effective but also I don't want to screw anything up.
As long as the second layer is done properly you won't "screw anything up". You will just have half your house with 2 layers.
As to what you save....
*the time it takes you to tear off half the roof
*the cost of additional underlayment and the time it takes to install.
*Maybe you get by with a smaller dumpster depending on how much roof your talking about.
Tearing the whole thing would be better but if cash flow is an issue then what you proposed falls under "do what you gotta do"