Sampling Rate and SNR
Your instincts are right. You won't increase SNR by increasing sampling rate because you also have more noise samples. The overall signal energy and noise energy does not depend on sampling rate.

But, I believe you are mixing up two problems. In your case the frequency is not constant, therefore Fourier based analysis yield less than optimal results. Your problem is in the signal itself, not the signal to noise ratio. You can have a look at the high-order ambiguity function (try this or that) or the Wigner-Ville transform. Those are more suitable for linearly changing frequency.