I am 33 & hope to have children. I don't have a partner, should I think about freezing my eggs?

Good idea. While freezing eggs will assure you of some percentage of viable eggs in time of future need, hopefully your search for partner will bear "fruit" sooner fir you not needing to resort to such backup plans. Good luck!
May be. Have you considered artificial insemination from an anonymous donor? Freezing eggs may work, but it is not certain, not to mention the expense. Using frozen eggs will necessitate in-vitro fertilization and its attendant issue. You may also consider adopting one or more babies. Wish you good health and healthy babies!