We have been engaged a long time and fear falling into sin. Can we marry first and hold the reception later?

Yes, and it is often better than prolonging the engagement.
A marriage contract that fulfils the pillars and is lawfully registered does not require a big ceremony. A couple can hold the contract at the religious affairs office or a mosque, then postpone the walimah until they can afford it. After the contract, they are lawfully husband and wife with all the rights and responsibilities, including maintenance, even if they may not yet live together because of studies or work. Discuss clearly beforehand: when to live together, maintenance while apart, and what happens if there is a divorce before living together. Never marry secretly or without registration. See walimah and maintenance (nafkah).