I have the same issue. We live in a valley, poorly covered by Verizon, or any other cell carrier. We have no direct line-of-sight path to any cell tower, and the nearest one is almost 5 miles away. I'm installing a yagi antenna on our roof and am in the process of trying to align it for optimal signal amplification using a Uniden U4 booster. Here are some things I've found out.
In the Field Test app, the "Base ID" identifies the tower with which your device is communicating. We talked to a helpful Verizon tech support person who confirmed this.
Cell phone RF, like light, is best served line-of-sight. In the absence of line-of-sight access one has to work with reflections and refractions from the surrounding topography, and trial and error is probably the best way to go. Getting to the mount of the yagi antenna here involves climbing a ladder to the roof and then a ladder on the roof to the mounting on a bough of a large oak tree - not a huge task, but proper alignment will involve a lot of up and down trips. I'm 72 years old and my wife gets very nervous when I'm working up in a tree!
cellreception.comis your friend as far as locating towers. I believe their data is crowdsourced since the major cell carriers don't make the locations of their towers available to the public. A useful technique is to identify likely towers on cellreception.com, pin them on Google Earth, where you can actually see them, and draw a line between your location and the tower. Go to a location along this line, closer to the tower where you can actually see it and check the Base ID. If it matches what you see at your home location then you've identified the tower which serves you.
Carriers pass off signal between towers based on a number of criteria including traffic volume, signal quality and even legal reasons related to coverage areas. So it's logical that your Base ID will change even when standing in the same place, or be different depending on the time of day. The tower that handles your traffic at any given time may not be the closest one.
A database correlating Base IDs with tower locations would be extremely useful, but the carriers apparently keep this information very close to their chests. The Verizon tech support person to whom we talked was willing to look up individual Base IDs for us, although the one we gave her was a 4-digit number which I would guess belonged to a Verizon Network Extender in the neighborhood; she couldn't identify it.