The thing to keep in mind with thermometers is that most of em have a tolerance range, and most of those ranges are pretty wide. And in cheaper thermometers, the range tends to get wider. Not a big deal for chilling, or maybe even fermentation, but for mash temps, can be a pretty big deal. I had a thermometer that was accurate in ice water, and accurate in boiling water, but not accurate in mash range. Off by about 5 degrees, which is a huge difference.
I highly recommend tossing out the "don't want to pay a lot" and buy a Thermapen. Fast AND very very accurate, and while ~$100 might seem steep, it's worth every single penny. That was one of the best investments I've made. The problem with probes is that you're only reading one spot anyway, not throughout the mash where you can certainly get pockets that are different temps. Unless you've got a recirculating system, you can't guarantee an accurate reading without stirring anyway. So I don't waste my time with em. The Thermapen is all I need.
As far as your ice water test, unless you did it properly (more ice in the glass then water and stirring constantly) it's not surprising you'd read high, because it's not a constant temp otherwise, and if you're just going based on a glass of water with 2 or 3 ice cubes it would probably never reach 32F before the ice melted entirely. That said, 80F is pretty high.