Do I need to change my travel plans if I don't get all my shots in time?

No. Immunizations are given to protect you from biological hazards you might encounter in your travels. If you fail to get all the recommended ones prior to travel, you incur a little extra risk. So the answer is, are you willing to assume that risk? If so, go ahead and travel. If it makes you nervous, then postpone and finish the shot sequence.
Yellow fever..... In most cases, it's ok- just be cautious with the yellow fever vaccine. In certain instances, people have been denied transit to a country if he/she has visited a "yellow fever" country. Typically, one needs a who- yellow fever vaccination card, issued by the clinic who administers it. In certain cases, exceptions are allowed. Regardless, i encourage you to visit either the who or cdc websites.