How are LFSRs used in real applications as PRNGs?
Sequential numbers are highly correlated, which is to be expected, given that if the output bit is 0 the next number will be half of the previous one. For a 15-bit LFSR with taps [15, 14], plotting a pair of sequential numbers as points in a plane gives the following. An ideal PRNG should spread these points all over the place.


If you want to generate random numbers with a 15-bit LFSR, you don't pull a new random number each clock cycle. As you said since you're only adding one new bit to the register each clock cycle, the value in cycle N and N+1 will be very strongly correlated. If you want to generate random values (assuming you have proper taps), then you need to only pull a new value every 15 clocks. 

A LFSR only guarantees you one random bit each cycle, not 15 random bits.