Does NASA actually have adequate funds to support multiple deep space missions?
To support them, yes. To launch them and build them probably not. Its worth mentioning that even cheap missions to another planet are in the tens or hundreds of millions of dollars.
One of the real problems with deep space missions though is that 1. We're running out of plutonium, which we need as a power sourve for missions beyond the asteroid belt, and 2. The current political climate is no conducive to the kind of goodwill gestures that allowed us to create START exemptions previously.
Launching nuclear material, which is currently a necessity for deep space missions, is forbidden by treaty. Exceptions were made previously, but the last one that was given out went to the ESA.
So first we would need to start making more plutonium, then we would need to have a launch vehicle for deep space insertions (pending, by SpaceX), then we would need to fund research 1 universities (rather than close them like we are trying to do now) to create deep space missions again. Last we would need to create an international accord for research of this type with the major players involved.
Under the last several Republican congresses however NASA has been slashed to the bone and beyond, because of its support for evidence based reasoning and climate change research. So if you want to know more about our solar system and beyond, make sure you vote, and vote Democratic.