Your situation sounds similar to mine, RedNeckGeek.
Perhaps you should consider a separate service using a router rather than a cell phone plan. It looks like Verizon has them:
Verizon 4G LTE Broadband Router with Voice | Verizon Wireless
I use a Rogers ZTE MF275R for my internet service (in BC, Canada). I am 30 km from the cell tower. The terrain
is good but trees close to my location have some effect. I changed from a different supplier last summer, mainly because of a much better data plan, but also to upgrade to 4G LTE technology that became available since the first installation a few years ago. I continued to use the Wilson 304411 antenna for a while and it gave reliable service with LTE download speeds of about 10 Mbps.
Service degraded in December. We eventually determined it was a Rogers problem. But in discussing it with the 2nd level support person I decided to upgrade my antenna to ensure reliable service and to take full advantage of LTE. I went with a YagiRef2-QLP antenna from WirEng.
To take full advantage of LTE you need two antennas installed at +45 degrees and -45 degrees. Here is a YouTube link to show why:
LTE 4G Part 2 - Why you must have 2 antennas for LTE speed - YouTube
There are other videos in the same series that give more information.
I bought the YagiRef2-QLP from BestCellDist.com. I believe they are based in San Diego but the antennas are shipped directly from WirEng in Texas. They were very helpful. I got them to send me an antenna user manual before I bought the antenna.
Take a look at the BestCellDist.com site and the WirEng.com site for good information on antennas.
The antennas significantly improved my signal. With just my ZTE MF275R's standard antenna the signal showed as -106 dBm which is marginal but gave a download speed of 10 Mbps. With the old single Wilson 304411 the signal was -100 dBm. Much better but the download speed was still 10 Mbps. With the WirEng YagiRef2-QLP it went to -92 dBm (which is 5 out of 5 bars) and the download speed doubled to 20 Mbps.
My installation required dual 50 foot cables. It is best to use ultra low loss cable for long runs. I used LMR400. The antenna has FME connectors and BestCellDist's YagiRef2-QLP includes pigtails to connect FME to F. My LMR400 have the small SMA cables on the other end so I could drill a smaller hole through the house wall. I used a short SMA female to SMA male pigtail cable to make it easer to connect to the router's SMA connector. I got my cable from InfiniteCables.com:
Antenna Cables & Accessories - Infinite Cables
You can get them at about 1/5 the price on EBay if you don't mind waiting 6 weeks.
I like the antenna and router for my service because it is less complicated that a booster. The downside is it only helps the router and does nothing for normal cell service (which I don't need).
Regarding the weatherproof connection from RG11 to F-Female: could you not just tape the connections and use a sealant? I did that with all my outside connectors.
Here are a couple of photos of my installation:
