It is definitely possible that you can become ill simply from moving from one city to another. Moving to a new place exposes you to new and different environments, climates, people, and living situations. These are all likely to be risk factors in causing you to become ill. Aside from that, the simple act of moving can be a very stressful time, which in itself can lead to you have a lowered immune system and have a higher risk of falling ill. The environment may be different, and smog or pollen levels may be higher in certain areas. Your body's reaction to these elements may make you feel ill or be triggering your allergies thereby giving you a feeling of having a head cold. Try to notice when these symptoms are worse and what environment you are at those times, it may be that in certain older dwellings these symptoms are exacerbated. This would be another sign that you are suffering from allergies. Of course, it is important to visit your primary care doctor
if your symptoms don't improve or worsen. They also know your medical history and clinical presentation and would be able to best diagnose and treat the underlying issues.