There are two things you need to look for, worn struts (yes, they are struts on all 4 corners) and warn out bushings. My car, which has 95,000 miles on it, had both. The suspension is definitely a weak point in the first generation Avalons, which share suspension components with the Camry, despite being considerably heavier.
If the shocks are worn out the car will float around too much and is more likely to bottom over bumps. The rattle you hear, can come from the shocks, especially the front mount, but is more likely coming from the sway arm bushings. The bushings are the cheapest to replace, other than the labor costs. The suspension arm ball joints can also wear out. You need to check those for play. You should also confirm the shock mounts are tightened. Sometimes they come loose. If you don't know how to do this yourself it can be expensive. When I first got the car I let a shop "fix" the rattle. They replaced the suspension bushings and bearings and it ran me almost $1000 at least half of it labor. I subsequently replaced the struts and springs myself. I wish I had done the suspension bushings and bearings myself but I was busy at the time.
On my car everything that could be bad was. All the bushings were worn out and needed to be replaced. The shocks were shot. The mounts were loose in the back. The rear springs had sagged 2 inches and the car constantly bottomed on everything.
The suspension wear is directly related to how hard the suspension has worked. The guy I bought the car from lived in an area with very bad roads and he wore it out driving over all the potholes.
We put Tokico struts ($350) and Vogtland lowering springs ($200) on the car because we wanted to improve it's handling. If your springs are OK, and you are OK with the ride, it is much cheaper to go with something like Monroe struts and keep the original springs. KYB is another one that gets good reviews and is a bit cheaper than Tokico, but more expensive than Monroe.
I would never let Sears or a similar place replace my suspension. They have a horrible reputation for auto work, as do many of the franchise auto shops. It is much better to talk to friends who have older cars and get a recommendation on a good independent shop or mechanic.
Here it is with 17" 215 50 series wheels and tires.
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