If you're actually supporting, especially if you have stuns, you should try and roll around with you carry/cores to try and and not die and land your carry some kills. If there's space, you can pick up some cs safely if you can (make sure you have a TP, or 3). Key thing is always pay attention to the map. Try and figure out what your team is doing and play with them. If it's 5v5 then figure out if you want to gank someone, smoke, farm, or be ballsy and rush down a T2 before they TP in. If it's 5v4 (your favor), maybe roll at a 5 man and rip down a tower and maybe another kill. If it's 4v5, then just play it safe and take careful farm and keep your wards up.
Biggest thing is your positioning and map awareness. If you're out wandering by yourself taking towers alone as a support and your team is dying in teamfights, someone is coming for you next, asap as well. If you're alone and they aren't on the map and you're in "their zone", you're out of position and you need to boogie, by foot or by TP, outa there and do something else.
I, personally, am also completely objective/tower focused. As such, I can recommend Shadow Shaman as a 4/5 and Pugna as a 3/4 so you can actually be useful if you're gonna take towers. Shadow Shaman is wonderful, especially with aether lens now. You can shackle someone and let your carry eat them in lane, if they aren't done yet then zap then hex them. If you've got your ult and they're dead you can push the wave, drop your ult on the tower and do some TD.
I have learned to love hard-pushing dota, which often turns into rat dota because I like to 1) not die, b) eat towers. Favorite heroes to do that these days are Terrorblade, Lycan, and Lone Druid. I announce I will either jungle or will lane if someone will support me in lane. This gives team mates the option to go solo-offlane core heroes (which are popular) or to be support and I'll thank them and be happy for faster farm and some kill potential. I plan to jungle all of them with a quelling blade, stout shield, tangos as a default though. All of those heroes I am comfortable playing in offlane, mid, safelane, or jungle-with some being harder than others. Surprising for some, TB works decent as a mid with reflection and meta these days in my neanderthal league (actually everyone's pretty nice, and I don't know my MMR).
As TB/lycan/LD I get my essential items as fast as possible and start taking towers as soon as I get what I need to rip one out and there's a bit of space. For TB that's Yash/Meta/conjure2-3, but prefer SnY first but it depends on the space. As LD all I want is Lvl4 spirit bear (with demolish) and I let the lane know what's up and we'll rip down the T1 quickly. As Lycan will push towers whenever there's an opportunity to rush in and bang on one if I see at least 3 enemy heroes elsewhere. Then run in with wolves (hopefully NecroX and Vlads maybe) and do some damage and peace out when you see the TP in. Ult if you need to panic escape.
One I'm comfortable with my farm/lvl/items (have one of my big ones) I'll rush BoT. BoT let you push a lane into a tower, run into trees, TP to the creep wave with nobody in it on the far side of the map and push in the creep wave there. If I feel uneasy, I'll run away. If it feels good I'll push in to the tower, maybe take it if I've got some buddies. Otherwise, TP back to another lane that's empty and push it into the tower. Your BoT have a 45s cooldown, their TP scrolls are 70s. If you keep pressure, they can't TP to you like half the time because it's on cooldown. Then you push the tower, bait the TP, run away and TP to the lane they just TP'd in from. I always keep an eye on the map though. If it looks like we need to run as a group, or I see them begin to look like they're going to engage, I'll TP back to my team preemptively to be there if they need me. If they don't, I'll wipe the jungle then get back to work. If they initiate like the first time I tried to ask a crush out, get the fuck out and watch that SK cancel his channel and blink in and walk around a bit before melting from afar. If you initiate or they initiate decently, go balls in, but try not to die and be a bro to your teammates. I love rat dota. I hate playing against rat dota. Like NP pisses me off. But I'm sure my 250dmg/hit TB illusions continuously pushing lanes into towers makes them made too. Or backdoor ult lycan tower ripdown with an AC/Necro3.....
Mostly I guess, watch the minimap and see where your buddies are. Learn what "out of position" is. Eventually you'll get this spooky feeling if you've had your eyes on the minimap and haven't seen anyone in a bit and you're a little too deep in. At first you're feel the spooky then instantly die because they blink/shadow blade/smoked/ran at you. Eventually you'll feel it, attempt to run away, then get melted. Then will come the "Uhhhh.....I shouldn't be here" where you run off then see three dots descend on the creepwave where you were at 2s ago. And life is good. :)