I have not yet witnessed a PRNG that didn't fail any of PractRand's tests after restarting its cycle, so most likely you are miscounting or misunderstanding how much data your generator should be able to produce.
Since 38 is neither a power of 2 nor a multiple of 8, I assume it to be miscounting.
An ideal generator with 38 bits of permanent state will produce 2^38 bytes only if each output is only one byte long. If, instead, each output is as long as the entire state, the full output will be 2^38×(38÷8)÷1024^3 = 1216 GiB.
Similarly, a 32-bit generator should output at most 2^32×(32÷8)÷1024^3 = 16 GiB before failing tests, and this is exactly what I've found when testing my own generators, as well as scaled-down versions of other generators.
If you're able to produce 2^38 bytes within a tolerable time, it should be easy to confirm whether or not the generator restarts after that point by simply comparing the next output against the very first output.
I can't give a better answer than this without seeing the code of the PRNGs.