If you have checked for and not found any rotor fasteners that hold it
to the hub, such as a large flat head screw, then it is probably just
stuck due to corrosion on the studs and the contact point (backside of
the rotor) where the rotor mates with the hub. Try using penetrating oil
around the studs and then use light blows with a dead blow hammer or
slide hammer to break it loose, or if you don't have either of those, a
board against the backside of the rotor using a regular hammer,
preferably a maul. Leave one wheel nut on a stud threaded on a bit, so
the rotor does not fly off and possibly hurt you as it breaks free
suddenly when you hit it. If the penetrating oil / hammer does not work
(rarely) then try heating with a torch around the studs and then the hammer.

You have not received very good advice for your problem.

Your rotor is not held on by anything...except rust.

The rust is NOT where the hub meets the rotor at the bolts, it is on the
/perimeter/ of the hub.

Rust in the "top hat" portion of the rotor is hooking the rotor on to
the hub. You must break this rust up in order to shift the rotor.

Penetrating oil will do absolutely nothing except make a hell of a mess.

After raising the car, securing it and removing the road wheel, do this:
1) Get an 8oz ball peen hammer. That's a half-pounder; a little guy.
2) Remove caliper and its mount bracket.
3) With ball end of hammer, tap on the side of the "top hat", about as
hard as you'd vigorously knock on a door. Maybe a bit harder
4) Tap as close to the friction surface as you can, to make sure your
blows are as far away from the hub as possible. You will damage the
friction surface with occasional misaimed whacks, but you're replacing
the rotor anyway.
5) Turn the rotor and keep tapping. Turn and tap, working your way
around the "top hat".
6) Eventually (a minute or less), you'll start to hear and see rust
breaking up and falling out the bottom of the rotor.
7) Keep tapping and turning.
8) At some point, you should be able to make the rotor move by wiggling
it with your hands. Find out where it's still stuck, and concentrate
your hammering there.
9) Once it's loose enough, THEN you hit the rotor from the back, to
drive it off the hub.

Sounds more complex than it is. Should take no more than 3 minutes per
rotor, with easy effort.

Warning: Stock up on 50-grit emery cloth (NOT sandpaper). You MUST sand
off every bit of rust from the hub, making it flat as Kansas. Failure to
do this will induce brake vibration with your new rotors.

The Avalon is basically an enlarged Camry, so the it should use the V6
version of the hardware. The rotor can move around the hub and wheel
studs normally. Tegger has some excellent suggestions there.

No rust with mine when I swapped out the OEM warped rotors with
Raybestos Raymold noise dampening rotors (red box) for about $20 each.
Akebono ProAct brake pads (OEM Toyota) are quiet even without the
noise shims on mine.

www.akebonobrakes.com
www.raybestos.com