Others may have some alternate suggestions, I'd get a feel for everyone's opinion before jumping in. I don't have a fogger yet

so I don't have any experience with the actual electronics involved, however...
I wouldn't use that relay, the amperage rating is the amount of current that can flow through the contacts (1 Amp on that relay), not how much current the coil requires to operate. I think it's too small for this. Look at this one instead:
http://www.radioshack.com/product/index.js...rentPage=family
The contacts on this one are rated for 10 Amps at 125Volts which should be more than enough for your fogger. To be sure, you can look at the switch. It probably has the specs printed on the side and as long as the current is under 10 amps this relay should work.
On relays, be careful not to confuse the voltage and current that the coil requires with the voltage and current that the relay contacts can switch.
As an example:
THE "INPUT" SIDE OF THE RELAY
The relay above requires 12 Volts DC and 30miliamps to operate. That’s what it takes to make it switch on/off. You mentioned that your wall wart can deliver 1000 milliamps (1 amp) so that is more than enough as long as the wall wart output voltage is 12 volts.
The voltage needs to match, and the supplier (wall wart) needs to capable of supplying at least as much current as the consumer (relay coil) needs. If it can supply more, that's ok. The consumer will only use what it needs. In this example your wall wart (gives 1000mA) could supply up to 33 relays (takes 30ma) with enough current.
Never supply more VOLTAGE than the relay needs though, if you do the best outcome is a shortened relay life, the worst case is a fire from the coil overheating and shorting out
THE "OUTPUT" SIDE OF THE RELAY
The relay above has contacts that are rated at 125Volts at 10 Amps. That's enough to switch on/off 12 100 watt light bulbs at maximum power. A 10 amp capacity may be overkill for the fogger, but a bigger "switch" never hurts. If you are only switching 5 amps of current on/off then theoretically the relay will never see more than half of its maximum load.
Think of it like a truck bed. You may be able to carry 3500 pounds of material in your bed, but it will also work fine for the 5 bags of mulch you need. You however would not want to balance those 5 bags of mulch on a motorcycle as it wasn't designed for that type of load.
DISCLAIMER:
I can't say enough to be careful working with these voltages, it's more than enough to kill you. If you are uncomfortable or unsure about exactly what to do, then don't do it. As a mater of fact, I'm recommending that you don't do it. I'm not a pro by any means so don't take what I'm suggesting as the gospel. It will work however as I and others use this method frequently with success (I have also let the smoke out of several pieces of electronics in my life.) Bottom line; you can't be too careful.
HTH, be sure to post what you end up doing.
Terry