Is there a right way?
I haven’t attempted this in decades, but now some motor driven something in the freezer has begun beating itself to death, presumably on the accumulated ice.
My inclination is to just toss everything that won’t survive a few hours in an ice chest, or is older than my nephew, turn it off, and…what?
Ice excavation; what’s the best method?
If you have a motor driven something in your freezer it is probably a frost free freezer which would mean that your problem is other than ice buildup. Is there visible ice buildup in the freezer?
Yes, lots of it.
If it is just ice buildup, call your friends and have a party with the contents. Otherwise just do as you said and put everything in a cooler, turn off the refrigerator, heat up a pan of water and stick it in the freezer compartment with the door shut. You may have to do it more than once if there is a serious amount of ice. Then just sponge out the water and turn it back on.
DON’T use a knife or screwdriver or any sharp object to speed things along unless you have enough money in your wallet to buy a new refer.
Prop the door open, turn it off and wait for the ice to melt/soften then scape and wipe dry. I’ve heard you can use hot (not boiling!) water or a hari drier to speed up the process, but the ol’ wait, scape and wipe method works for me.
Mind you, I’ve only had the chance to do this twice in my life.
I second using a hairdryer also a rubber mallet to break free large ice chunks. I don’t know how old your refrigerator is but it may have it’s coils exposed be careful not to damage them.
OK, sharp objects warning noted (I was going at it with a knife, but hadn’t reached any refrigerator parts yet).
Famous last words.
HISSSSSSSSSS
Heh. I don’t know about HISSSSSSSSSS, but the motor driven something has calmed down considerably.
Remember, when it’s time to turn it back on, that fridges have built-in cycles. They will not run continuously. So if it’s in the off cycle when unplugged, it may take 20-30 minutes before it kicks in again, no matter what the cold-dial is set at.
Empty the drip pan as you go or it will be so full you can’t get it to the sink without spilling most of it.
Also, there is always ice between the walls. (They put tiny air holes so it won’t smell stale when the unit is sitting dry in the store.) When this ice melts, it often goes down the insides and misses the drip pan and ends up on the floor. So have old towels lying about.
Go ahead and use tools to chip the ice. Just do it gently.
An old butterknife works better than a screwdriver, and can slip into cracks. A light tap with a hammer can break off big chunks.
When the dam breaks, it’s handy to have sponges to get the water out of the flat areas.
The noise was likely accumulated frost impinging on the fan impeller. Assuming a full size refrigerator/freezer, if it was manufactured since the early seventies, it is likely of the frost-free mode.
Most frost-free units work on a very simple principle: a clock timer turns the refrigeration compressor off, and the defrost heating element on for a given percentage of time, such as 10 minutes every 24 hours. This is usually sufficient to cause the mild accumulation of frost to melt, and the moisture runs down a tube and into a pan beneath the compressor. At the end of the defrost cycle, the compressor restarts, and the radiant heat from it’s operation evaporates the water in the drip pan.
If the cycle timer has died, then any moisture entering the cabinet will freeze on the evaporator coils, and the unit will run longer and longer, as the ice impedes speedy heat exchange between the air and the evaporator coils.
Defrost timers are typically under $25, and can be replaced in about 5 minutes using a screw or nut driver. Every unit I’ve ever serviced has a molded plug and two or three screws securing the timer to the cabinet or frame. They’re usually about the size of a pack of smokes. The trick is finding the blasted thing.
If you’ve got an appliance repair house near you that will sell to Joe Public, take your model and serial numbers along and inquire about the cost of the timer. If the counter person is nice, and you ask politely, they probably have an exploded pictorial diagram showing you where the timer lives. Once you’ve found the timer, if you can unbolt it and leave it connected electrically, feel the timer or listen to it. Even the quietest ones make a soft whirring noise that you can often feel. No sound at all=bad timer.
Hope this helps. 
The fan may make a lot of noise for either of two reasons:
the blades are coated with lumps of hardened dustbunnies.
This makes the fan inefficient, as the airflow is not smooth and thus less air is moving. Also, the balance will create the noise and eventually ruin the bearings.
These lumps can be rinsed off, but gently as the blades warp easily.
the bearings are shot. A fan motor may run $50. And be sure to get the blade if needed, as these are sold separately.
Also, while the machine is off be sure to clean the dust off the coils under or behind the unit. This is the major reason that a unit will seem to be not working. It’s working fine, but the dust is insulating the heat right where it’s supposed to transfer to the air.
A coil brush is just a long bottle brush, and if you can’t reach far enough with other things, you should get one at Ace or TrueValue hardware for $5
If it is a frost free refrigerator it could be a timer or other component in the frost free circuit. If so a technician is in order when all else fails and a substantial bill.
Pans of hot tap water (you can put your hand in) are OK
Easy on the metal tools, hamme is not reccommended.
Helpful Household Hint - After you’re done with the defrosting and the freezer is empty and dry, spray it with Pam or some other cooking spray. Then, next time you need to defrost, the ice will come off in sheets and make the whole job a lot faster and easier.