Distribution based integer factorization
The archetypal algorithms that could take advantage of $p$ being close to the square root of $N=pq$ is Fermat's factorization method; and some improvements thereof.

The basic Fermat factoring method makes about $(p+q)/2-\sqrt N$ steps. Applied to $N\approx 2^{127}$ and $p/\sqrt N\approx 0.98$ (as in the question) this is over $2^{51}$ steps. We have heuristic algorithms to factor any 128-bit RSA moduli much faster than that, and nearly as simple (e.g. Pollard's Rho), thus the method is worthless in practice even at the 128-bit level; that only gets worse when we increase the modulus to practical sizes, and compare to more powerful factorization algorithms.

There are improvements of the Fermat factoring method that still benefit from $p$ being close to the square root of $N$. They reduce the number of steps, by skipping most candidates, and performing table lookups rather than squareness tests. The gain is large, but severely bounded by available memory, and is not enough to become competitive at least for $N&gt;2^{1023}$ and $|p-q|&gt;2^{\lceil\log_2N\rceil/2-100}$ (the criteria in FIPS 186-4, with origin in ANSI X9.31:1988, that has plenty of headroom), which has overwhelming odds to be verified for random $p$ and $q$.