STFT on time varying signal with good time and frequency resolution
I think that you are splitting your signal $x[n]$ into $N$ STFT intervals which you are overlapping by 50%, then on each interval finding the Power Spectral Density (PSD) with Welch's Method. 

If this is correct I think this method is valid, 

but I think you might be confused by the 'overlapping' because there are two at play here. When you use Welch's method you are splitting the STFT interval into smaller intervals, from which the periodograms is calculated, and then these are averaged to give you your PSD for that STFT interval. 

So there are some variables which can be changed here


The length of your STFT intervals, $N$
The overlap of your STFT intervals 
The overlap of your welch's periodograms


The smaller the length of the STFT the less smooth your 'time resolution' will be because you are averaging over a smaller length of time which will allow you to pick out sudden changes in frequencies, but less data is obviously going to have some kind of negative effect (maybe lower frequency resolution, but im not sure)

I think how much you overlap the STFT is quite related to the 'time resolution' here because you can think of it like, for the first interval you've windowed you are finding the PSD over $[x[0] - x[N]]$ but then you could quite easily take the next interval to be $[x[1] - x[N+1]]$ these two clearly overlap depending on $N$, but its better to just think of you moving your STFT window along $x[n]$ and calculating the PSD at each one, so in this sense your 'time resolution' can just be the same as your sampling rate, but its smoothed by the overlapping.  

The overlapping of the periodograms smooths the frequency resolution but also reduces noise

I'm not sure what Jains method is but if you wanted to look at higher frequency resolution methods I would look at instantaneous frequencies using the Hilbert-Huang Transform, see this paper for some comparisons