to the OP, you will 100% not have to go through security again at FLL. The Silver Airlines planes (UX) park at gates C5 and C8. United uses the majority of the rest of the C Concourse.

For reference for others using FLL, Concourse B was still safe as of March. The others usually have at least one lane open that the Backscatters are turned off.

But back on topic, you are 100% safe connecting UA-UX and vice versa at FLL.