UX designer here. See what kind of work and industry interests you - in-house product, agency, enterprise... then apply to all entry level positions that look interesting. Internships can be a good way to get a taste of position before fully committing, but their availability and quality really depend on location.
As both options are entry level, recruiters expect little concrete projects in portfolio (few personal projects should be ok), they will look for potential to learn how to do the work, so I suggest highlighting your ability to learn, think in a structured way, challenge your own assumptions, work with other people and grow.
Good luck and welcome to the world of UX design!