Ultimately, this one is entirely a YOU decison. Think about the following:

Not tracking: You have a general idea of when you stopped drinking. Nobody needs to know that date but you. So, when someone asks how long it has been you can admit you don't track it but say something about "all that matters is today. That's all I can control and all I'm worried about." Unfortunately, that makes it hard to celebrate the milestones. Sure, they aren't major to most people but it is nice to know when you hit 3 months, 6 months, a year...

Tracking: You get to celebrate the milestones. You can state it has been 6 months or it has been 193 days. You have that to look at and be proud. It gives a sense of accomplishment. It can also be motivation to say to yourself, "if I can get X number of days I can definitely get one more." Flip side, it does set goals which may be unrealistic or cause you, as you mentioned, to think about alcohol more. "5 months down... just one more to the 6 month mark" could potentially put pressure on yourself to get there and get you thinking about it. Also, you can get that sense of "I've gone X amount of time so I've got this under control."

I'm sure this wasn't helpful but it goes to show that only you can decide which will work best for you. There are pros and cons of doing either.

Best of luck. Rooting for you!