Hi! I'll also be starting my PhD in the US as an international student, but I've moved abroad before on my own so I might be able to help.
Firstly, no one will think it's weird if you arrive with 3 giant suitcases, especially since you're an international student!
You could consider taking the clothes you're likely to need in the first few months, and have the rest shipped over later? Since you're committing to a long time in another country you're likely to want quite a lot of things from home with you anyway.
In terms of towels, etc. I'd recommend taking just a travel towel so you have immediate access to one, and buying more later since they're bulky. I'd take a coat as hand luggage to save space. Only take a few travel toiletries and buy the rest when you arrive - things like that really do take up space.
Having said that, it really is normal to have a lot of luggage when you move, especially when you're moving abroad, so I wouldn't worry!