If someone wakes up and touches another object before washing his/her hands, does that object get some level of tumah as well?
Simple answer no.
Complex answer, there is a mahloket as to exactly why we wash our hands in the morning. Some hold that is is because we have touched an unclean part of the body, some hold that is is because of a ruah ra that rests upon a person while they are asleep, some hold that is not a ruah ra but rather klipot or hiztoniut that rest upon a person while they are asleep, and thus must be driven off before one can start to pray. However none of those things is listed by Hazal as a form of tumah that is capable of being passed on, to an additional object.