Yes Cockroaches can and do spread diseases especially when they crawl over any food or dishes.
If the Cockroaches have crawled over your food you need to throw the food out because it could have diseases in it from the cockroaches and make sure you wash your dishes in really hot water.
Do your best to keep your house as clean as possible and don't leave food out on the counter or anywhere and always wash your dishes as soon as possible to avoid cockroaches coming into your home.
If there's no food for the cockroaches to feed off of they'll find another place to eat.
If you can afford to get your house cleaned up and get an exterminator in your house soon to get rid of those cockroaches for good.