Your best tactic is to avoid. 3, 4, or even 5 quotes is enough to give you an idea of what the price will be. Remember that carriers set the price, not brokers; the only rate they control is their own fee, which is pretty much the same everywhere. Your main criteria should be trustworthiness, not price. Look at reviews and look at this post. These companies will not bombard you with phone calls, will get you an accurate price, and will find a reputable carrier.
If you already contacted a mob of brokers but do not need any more quotes, you have to say you moved the car already. Do not ignore the calls, they will keep calling. Even if you decided not ship your car, pick up every phone call and tell them that the car is long gone. Within a week or so the majority of the calls should cease.
If you insist on getting 20 quotes, do yourself a favor and set up a separate email and, if possible, a separate phone. Try to avoid giving your number where possible. Do not put a fake number if you do not know where it goes, you could be deferring those phone calls to someone else. And try to find honest companies, even if price is your central concern.
Chances are the companies that’ll contact you a million times aren’t companies that you want to deal with anyway. It’s not a sure sign, but it’s a sign nonetheless.