Provided that OpenStreetMap data in your area contains house numbers (in not all places is this available), OSMAnd works fine for finding exact address. The only catch is that you need to choose the city, then choose the street, then choose the house number. This is exactly how car GPS units worked for many years, but I know people today have been spoiled by Google Maps and expect the address to be found if they just type it in as freeform text.
Searching for the nearest grocery store in OSMAnd is trivial: go to Search and then tap on the shopping basket icon: a list of nearby shops will appear, sorted based on how close they are to your current position. Of course, the grocery store needs to already been added to OpenStreetMap. If it isn't there yet, you can always sign for some an OSM editor account and add it yourself (but be aware that with great power comes great responsibility).
Searching for the nearest grocery store in OSMAnd is trivial: go to Search and then tap on the shopping basket icon: a list of nearby shops will appear, sorted based on how close they are to your current position. Of course, the grocery store needs to already been added to OpenStreetMap. If it isn't there yet, you can always sign for some an OSM editor account and add it yourself (but be aware that with great power comes great responsibility).