As far as I understand, numbers produced by a PRNG will only be random relative to other numbers from that same sequence of numbers, not relative to numbers from a different sequence (with a different seed).

Only true for crappy PRNG's.

If you want to generate random numbers from a sequence of seeds (not a sequence of random numbers from a singular seed), look into hash functions instead.

Sure, what you're looking for is the term "DRBG" (deterministic random bit generator) which can easily be built from hash functions, they tend to be a bit slow with hash functions though. Better to build them from block ciphers, it's more efficient.