We just got back from Cancun. We arrived on 7/19 7pm CT t'was quick, smooth and uneventful. Departure was completely different such that there were some people held at the counter with problems like yours where the name on their ticket didn't match their passport.

We booked with United also and I remember the instructions to put our names exactly as it appears in our passports because they will not allow a change of name once the booking is confirmed. I suggest you contact the nearest Phil consulate for an update on your passport information and hopefully, you get your passport in time before you leave.

Feel free to PM me for details and what to avoid upon your arrival if your destination is Cancun.