The FE/PE licensing rules vary by state. Note that ABET accreditation is technically program-specific, not department-specific. In most engineering departments, only the BS programs are ABET-accredited, not the MS programs. So even if you get an MS degree in Ocean Engineering from a department with ABET accreditation, the MS degree may not technically be ABET accredited. It looks like the only true ABET MS in Ocean Engineering is the one at UHawaii.
Some states may be OK with an MS from a department that has ABET-accredited BS programs. Other states may strictly require an ABET degree, so an MS degree may not work unless it is from UHawaii.
The FE/PE licensing rules widely by state. Some states strictly require an ABET degree. In others, you can qualify for the licensing exams without ABET degrees, if you have sufficient experience, but the amount varies. An ABET degree is always the fastest track.
This may vary by state. Some state boards may be willing to evaluate non-ABET degrees. Others may not.
I am not an expert in this field, but in general FE/PE is important for permanent, site-specific infrastructure and less important for portable devices. My guess is that if you want to design coastal infrastructure, you will probably need a PE license. If you want to design ships, then maybe you don’t.
If you want to design coastal infrastructure, then you might want to start with an ABET-accredited Civil Engineering BS (which your state university probably does offer), and then add an MS in Ocean Engineering. If you are in a coastal state, the Civil department may well have some electives related to coastal engineering. You would probably not have a lot of room for oceanography electives in a Civil BS program, although you could maybe double-major if you opted for a fifth year of undergraduate study.
If at all possible, try to find some actual ocean engineers and discuss with them. If they are licensed, check what licensing exam they took, because there is no Ocean Engineering PE exam. You would probably have to take either the Civil PE exam (which is offered by all states), or the Naval Architecture PE exam (which is less common, and my impression is that it is oriented towards ship design).