With a double axle trailer you most likely will not experience catastrophic sway or an out of control situation. But..and as Homer Simpson would say... "and it is a big but", a blowout at speed can and often does result in lots of damage to your trailer.
Tires are a key part when it comes to safety and are often overlooked. Tires that are cracked should be replaced. Good idea to have a professional look at them but my guess is that they will want to sell you some replacements. Get second and third opinions and you will probably hear the same story. This is a great example of "better to be safe than sorry".