Current OsmAnd~ version in F-Droid (i.e. 2.6.3) has a Legacy search option. To enable: Menu List > Plugins > OsmAnd development > Settings > Show legacy search.
When enabled, Legacy search is available from the menu list. Under tab Address it has fields for Region, City, Street, Building. For each field you can scroll through a list.
@olh, is this the search function you’re looking for? (I’m new to OsmAnd). This way of searching seems very useful in some situations.