Is there such a thing as a continuous phase BPSK? My knowledge of signals and signal processing is limited, so I apologize in advance for my naïve and possibly confusing question. 

In the case of BFSK, the frequencies are altered to represent the data (1's and 0's). The frequency transitions are abrupt. You can use Continuous Phase BFSK to "line up" the transitions in frequency. 

Now, for my question: lets say I am using BPSK which uses changes in phase to represent the data. If the phase changes from 0 to 180 (or vice versa), wouldn't that have a similar abruptness as BFSK? Is there something I can do, similar to CP BFSK to smooth the transition out?
You're right that the transitions can be abrupt, but you can always choose to use pulse shaping, i.e., instead of using a rectangular pulse, use some smoother function $p(t)$. The corresponding baseband signal is then

$$s(t)=\sum_kA_kp(t-kT)\tag{1}$$

where $A_k$ are the symbols, and $T$ is the symbol length. A smooth function $p(t)$ will avoid abrupt transitions, and will therefore reduce spectral sidelobe levels. I think this paper uses that idea.