Wow, I mean, they're pretty bendy so I'd be surprised if they totally snapped cause also subq tissue isn't exactly firm either. What makes you so sure it snapped?

Yes, it can happen that a piece of a needle can get into a vein - but you have to be aiming for a vein. You almost certainly didn't get it into a vein where it will travel to your heart/lungs/kidneys whatever because you were aiming for subq tissue and not a vein. It can happen with IV drug users (or in a hospital accident of course). This is pretty rare and it usually requires very dull very used needles or some really weird flailing or something. Believe it or not, there's people who do body mods where they purposefully leave things in subcutaneous tissue and it doesn't move to the heart. There was a group that did this with small magnets in their finger tips so they could feel magnetic fields! (The magnets were coated in a non biodegradable and body safe wrapping, I think gold).

Depending on your injection site, you might be able to flash a flashlight really really close to your skin and see what's there (you can see veins this way! It's neat). You could also measure the needle - the size should be on the packaging, measure the needle you have to how long the packaging says it should be. If you're really really scared, you can get it checked in an xray - I know it's not free but it's something.

One of the main dangers with foreign bodies in your veins is embolization, so if whatever limb you injected to is starting to hurt in a weird way or starting to get more puffy than the other, go check in emerg (it'll be obvious, not a "oh I don't know if it's happening" check out DVT photos and compare the affected leg to the non-affected leg). Your lungs tend to catch embolism and foreign bodies in veins rather than fucking up your heart or brain, you got a lot of mucous and a whole lot of proteins that'll grab on to the foreign body. If it's in your lungs, you will notice it will not be a "oh I dunno." You'll be breathing like 40+ resps a minute and feel pain. Now, the trouble is that panic attacks also feel like that 😭 so if you're getting especially anxious and have had panic attacks before, think a panic attack before an embolism.

You will be safe. It's okay. You have your gf, you have a good head on your shoulders, you know when and where to go for help, you know the warning signs, it'll be okay.