H-1B lottery/random selection

I am on OPT, which started on 19th February 2024. I joined my first job on OPT on 8th July 2024 and lost one attempt at the H1B lottery. I work for a research foundation, and they can sponsor me for an H1B-exempt visa. I have a degree in computer science, and I ultimately want to work in corporate America, which means I need to get an H1B cap-subject visa.

My question is whether I should get this H1B exempt visa through my current organization or if I should try to switch to a different company and use my remaining two attempts at the lottery. I ask this because once I get the H1B exempt visa and change my status from F1 to H1B, I won’t be able to switch to a for-profit organization and would be stuck in H1B exempt visa category. H1B-exempt visa jobs are less and pay less. And though the option of an H1B concurrent visa is there but I feel in reality, it will be challenging to get a company to allow me to work for another company due to NDA requirements and also allow part-time work. (But I would appreciate your advice on this since you must have had client experience.)

Can you advise me on how to get the H1B exempt visa now? Does it have any advantage, given I want to transition to a corporate company? What options will I have if I switch to a for-profit organization? Also, I am currently funded through a grant, and it has runway till July 2026.

I have seen people saying not to get an H1B exempt visa as you will be stuck in that category, but I also feel that any H1B visa will be better than F1.