Hi.
When you take a web maps offline, you do not get geocoding support from that web map. You would have to get hold of .loz/.loc locator files for the area in question (you could use ArcGIS Pro to create those if you have the source data, see Introduction to locators and Create a locator in the ArcGIS Pro docs).
Mobile Map Packages are another way to get geocoding offline, but again they are not derived from Web Maps, but created in ArcGIS Pro (which packages up the .loz/.loc files inside the .mmpk).
For an overview of the offline options that Runtime supports, see the Offline conceptual guide.
One thought: you say that the web map contains address search information. Can you provide more details? If you have a feature layer of addresses that's downloaded with your offline map, you can query that to find feature locations. It's not as optimized as building a locator from your data and doesn't support Suggest capabilities, but can work in a pinch if the data is right.
Hope this helps.
Nick.