How to generate safe primes in a verifiable way?
Here's a very simple method: Find the largest number below $2^n$ that is a safe prime.  Use standard primality tests for $p$ and $q = (p - 1)/2$.  For example, $2^{2048} - 1942289$ is the largest safe prime below $2^{2048}$.

But you didn't specify what you want this for.  If you want to use this with Diffie–Hellman to resist discrete logarithms, then that won't be a good option.  The bit lengths you describe are designed for DSA, which has different security requirements.

For an $n$-bit safe prime to use with Diffie–Hellman, you want few small subgroups and you want to destroy the structure that the SNFS exploits, so you can pick the smallest $c$ such that $$p = 2^n - 2^{n - 64} - 1 + 2^{64} (\lfloor 2^{n - 130} \pi \rfloor + c)$$ is a safe prime and congruent to 7 modulo 8.  The latter condition ensures that 2 generates the subgroup of quadratic residues in $(\mathbb Z/p\mathbb Z)^\times$, of prime order $(p - 1)/2$.  This is the technique used by RFC 3526 to pick standard groups at sizes from 1536 to 8192; the technique is described in RFC 2412, Appendix E.

This technique is sometimes called NUMS, for nothing-up-my-sleeves, because it uses the conventional transcendental constant $\pi$ instead of some inexplicable string of 1920 bits.  There's no security significance to $\pi$ except that it destroys some structure the SNFS could exploit—you could use $e$ instead, or $e^\pi$, or $\cos 1$, or all manner of other options to get a result you want if you knew of a small, say one in a million, fraction of primes that admitted a back door.  For this reason, may I interest you in doing Diffie–Hellman over rigidly selected elliptic-curve groups free of magic constants instead?  As a bonus, you get higher performance, smaller keys, easier defense against timing side channels, a number of high-quality implementations, and cooler names like X25519.

I'm not sure offhand what all the security requirements for Elgamal encryption: approximately nobody uses Elgamal encryption these days.  Naively, if the recipient given $(c, d)$ yields $d \cdot c^{-x}$ where $x$ is the secret exponent, then the adversary can apply the Lim–Lee active small-subgroup attack by supplying $d = 1$ and $c$ of small orders $n_0$, $n_1$, $n_2$, etc., to learn points ${g_0}^x$, ${g_1}^x$, etc., to which they can apply discrete logarithms in small subgroups to recover $x \bmod n_0$, $x \bmod n_1$, etc., and reconstruct $x$ with the Chinese remainder theorem.

Could I interest you in replacing your use of Elgamal encryption by X25519 in a NaCl crypto_box or libsodium crypto_box_seal, which have none of these finicky considerations and run much faster with fewer side channels and have smaller ciphertext expansion and are widely implemented and understood?