There are several ways you can handle this. The two ways you have mentioned, using a fan or a resistor in series with the motor, will both work but aren't necessarily best. They both work by limiting the total current that can be drawn from the power supply. But they limit it ALL the time instead of just on startup and they waste power when the motor is running.

Ideally what you want is to limit the current only during the time the motor is starting. There are different ways to do that. Perhaps the simplest is to use an "inrush limiting thermistor." A thermistor is a resistor that changes value depending on temperature. This specific type is designed to lower resistance when it heats up so that the initial surge of current heats it up then it lowers resistance. That is probably your best bet. Google for "inrush limiting thermistor" and you'll get notes on how to use them as well as places to buy them.

But, no matter what method you use one of the first things you need to determine is how much current the motor actually needs when running. You didn't mention what the motor is doing (what load it has) which is important. The amount of current it uses depends partly on the load. The motor may have a data plate that says, or you may be able to find specs online. Or, if you have a multimeter that measures about 10 amps you can just measure the current. You will need that information to determine ANY type of current limiting unless you just want to continue "cut and try" which I don't recommend.

A few notes on what you've done so far. The fan you used has a max current draw (usually listed on the fan) that applies when it gets its full 12V. Your fan + motor circuit can NEVER draw more current than that. (That's not quite true. The fan is a motor with its own inrush current. But it's likely smaller.) It is likely much lower than the max of the motor. In addition, the voltage then divides between the fan and the motor. You can measure that while its running. Let's say they divide evently, each getting six volts. Now your motor is running on 6V instead of 12. It is likely the motor is getting somewhat less than six. The resistor does the same thing but without an inrush current for the resistor.

If you want to use a plain resistor, a little Ohm's Law goes a long way. If you aren't familiar with Ohm's Law you should look it up. It is a foundation of electronics. Here is a brief overview:

I = V/R Current through a resistor (I) is Voltage (V) divided by the Resistance (R)

R = V/I Rearranged to find the resistance from the known voltage and current

V = I * R Rearranged again to find voltage from current and resistance

So let's say you want to limit the inrush current to 3 amps with a resistor. Use 12V and 3 A to get 12/3 = 4 Ohms. A four ohm resistor in series with the motor will make sure the circuit never draws more current than the power supply can handle. Resistors also have a power rating: how much power they can turn itno heat without damage. Common values for small resistors on circuit boards are 1/10 Watt, 1/8 Watt, or 1/4 Watt. You will most likely need one much larger. The full load on startup will be 3 amps and 12 V. Power is volts times amps. So that means 36 Watts. However, once it starts running the motor will take some voltage and limit the current even more. The resistor can take an overload for a short time while the motor starts. But you need to know what the steady power rating needs to be. Again, you need to find out how much current the motor draws.

This has been a lot of information and most likely confusing. If you have any questions, just ask.