Linear feedback shift register combination, secure?
I think Trivium, and ChaCha20 are excellent choices.

If you insist on using the shrinking generator or the self shrinking generator, you should use them without hiding the polynomials, which would be security by obscurity. I'd suggest that the SSG is probably stronger than the SG, for given keylength. But there are fast correlation and guess and determine attacks on both, so your real strength is only a fraction of the keylength. This would mean using at least 256 bit length LFSRs.

Finally they haven't been designed with modern Key/IV based operation. Key/IV based operation would help you achieve the goals you might have in mind for the question of whether you should change key and/or the secret polynomial.