I am not certain about these “time-chasers” you speak of. …
One thing I know from experience; it is difficult to keep up with your current ‘reality’. Events, prior to your current space-time, or forward, from whence you came, are in many ways (Some subtle and sometimes major) are actually quite different from your perceived memory.
It’s extremely important to mentally prepare yourself for unexpected changes, however insignificant they may be. Always thoroughly research current events and analyze carefully any ‘footprints’ you leave behind. Your grip on reality becomes a slippery slope if not careful.
Honestly, mental preparedness is the number 1 tool to safe time travel. Aside from that, there is nothing to fear, assuming you are properly trained.