yes, your solution sounds good… but you must check the antennas you put on site : what is the vertical beamwidth ?
if it is more than 8°, then a tilt of 2° or 4° should not affect the coverage so much (maybe a little bit).
if it is less than 8°, then yes, it can be the cause of the trouble.
As we said, if the poor HOSR is related to poor coverage, the only way to understand exactly what is going wrong is
1. to perform a drive-test
2. find where the traffic is making the most handovers (compare a map, the roads, the villages, with the time advance measurements)