Hi Patty,
You have several issues here and we'll just ignore the fact that you couldn't tell you had a flat tire

.
1. Why didn't the sensor work?
That's a weird one. Mine flashes if there's as little as a 2 pound difference side to side (discovered after a tire store rotated my tires and messed with pressures), yours certainly should have come on with a flat. That may be something for your Toyota dealer to check out 'or' when you're rolling again try an experiment because I'm thinking maybe the sensor wasn't properly calibrated to begin with (should have been done after each tire rotation)
Make sure all your tires are inflated to their proper pressure, re-calibrate the system by holding the button in for several seconds (until the tire light flashes) then, after you've driven awhile, let a few pounds of air out of one tire. By the time you've driven a mile or so the light 'should' come on. If it does air the tire back up and consider the earlier failure a fluke. If it doesn't it's dealer time. Don't forget to air the tire back up.
2. Need new tires
27,000 miles seems mighty early to need 4 new tires. Check them yourself. All tires have 'wear bars' built into the tread. When the tires are worn to a point where only 3/32" of tread is left the wear bars are even with the tire tread and it's replacement time. Another way to check (this is pre wear bar so I'm dating myself) is to use a penny. Stick the penny into the tread all across the tire. If you can still see Abe's hair anyplace on the tire it's replacement time.
Note, if your tires need replacement because they're worn in the center but not the outside of the tread it's because you've been running with the tires over inflated. If the outsides of the tires are worn more than the center your tires have been underinflated.
If you don't see any wear bars on your 3 remaining tires then maybe the tire store is suggesting replacement because they're now five years old (replacement is highly recommended between 5 and 8 years no matter how much tread is left), just trying to inflate profits 'or' they like all vehicles to have four matched tires.
3. What to buy
It's research time for you. The proper tire for your vehicle will usually come down to two things. Your budget and the type of driving you do. Obviously you don't drive a lot and may not need a performance tire. If that's the case you can probably get away with the cheapest (buy 3 get one free) tires sold at Pep Boys.
If I've guessed wrong and you really enjoy pretending you're at Sebring as you rip around the cloverleaf then by all means ask the dealer to show you a performance tire. The tread doesn't last as long but based on your current mileage and the cars age that isn't a big worry for you. The tire will grow old before the tread is gone.
And finally, I'm running Michelin's MXV4 on my '04 Solara Convertible. It's not the stickiest tire made but the performance is in line with my driving style/conditions, it 'is' a very quiet tire and seems to be wearing well. You'll find info here:
http://www.michelinman.com/tires/lux...rgy-mxv4-plus/
Cheers,
Gadgetjq
Host, Gadget's Solara Page
www.gadgetjq.com/solara