How can I travel outside the U.S. while I change from B-1 to F-1?

I’m an international student who recently graduated from college in the U.S. I entered the country on a B-1 visa, then changed it to an F-1 to study. I now have a full-time job and plan to apply for Optional Practical Training (OPT). But I am concerned about getting an F-1 visa for traveling abroad. Can my status change affect my chances of getting the F-1 visa stamp at a U.S. consulate abroad? Can I travel outside the U.S. during my OPT period? Does having an EAD card, a job offer letter, and my I-20 form endorsed for travel by my DSO support my F-1 application? Should I secure a long-term visa to travel more freely without the risk of re-entry issues, like an H-1B visa, or directly apply for an EB-5?