I often visit a friend who is in between some multi-site LASD systems and have the same issue. If you stay up in the wee hours of the morning, you'll hear the system doing automated phase testing with tones. At least that is what I was told last time I was there.
It is very possible that the LASD system has changed their multi-site coverage a few days after your new install, and your new antenna has directional properties that make it worse than the old one. If LASD hasn't changed their site setups, it is possible that you are getting some new multipath from some other source out there - new buildings etc which your new antenna is sensitive to.
A directional antenna might help attenuate one of the sites more than the other to make things tolerable. In the past, I've successfully homebrewed UHF "half-squares" very easily to try and put one of the sites in the azimuth null. It doesn't always work depending on the location of the sites, but the half-square vertical is easy to make for permanent or portable work - although you will get some stares.
The half-square vertical is a broadside antenna, with nulls along the ends. Basically it is just phasing two 1/4 waves together, and fed at one corner. Quick example: Cut 6-inches of say #12 house wire. Attach this to the center conductor of the coax. Now cut 18 inches of wire and attach that to the shield. 12 inches away from the shield, bend 6 inches of this 18 inch wire upwards so that you have the two vertical elements in line with each other - ie a half-square shape. You can use it with the wires hanging down, or the wires facing upwards - your choice. Rotate in azimuth to try and put one of the systems into the null. I've used broomsticks for the boom. For best results, try to run the coax feedline straight away from the antenna for at least a few feet.
Forgot to mention that broadside, you'll get about 3 db gain over a single quarter wave vertical. But I think that for LASD, you'll be more interested in the sharp nulls.
Reference: ARRL handbook, 20th edition, Page 6-17. While they describe using it for low HF bands, it still works at UHF.
This might be a cheap and simple antenna for your LASD monitoring - although depending on the system, you may have to rotate it.