Dave, first, welcome.
Second, clarify a bit. 'Edited and Saved image 0001'. Saved how? Lr doesn't Save, per se.
Here's how I'm interpreting what you said, correct or clarify as needed.
In the Develop module, with the filmstrip at the bottom of the screen visible, you make adjustments to image 0001. Subsequently, you Export image 0001 to a destination folder. Following that, you select/scroll to image 0002, which becomes the image in the main Develop display, and (most)selected/highlighted image in the filmstrip. You then see the filmstrip thumbnail of image 0001 change? If you scroll back to image 0001, it's different from the treatment you exported 10 seconds ago?
As you describe it, this is quite peculiar (as you've obviously noticed). That said, there are some features within Lr, that could possibly cause unintended changes, but we'll need to more info to diagnose. A detailed step by step description would be very helpful. Lr also has its own persnickety vocabulary, so we may be a bit pedantic, but it's all in the service of a clear verbalization of what's fundamentally a visual experience.
My first guess would be that you have Auto-Sync accidentally enabled in the Develop module, with multiple images selected. Have a look a the buttons, lower right hand Develop panel, and see what they say.