Can I get an std from sharing drinks with someone who had blood in their mouth or has given oral sex?

Here are some... People contract infection at exposing to causative bacteria if the natural barrier - skin &/ or mucosa is broken. Active bleeding indicated such breaking and oral sex may induce invisible micro-injury, leading to contracting infection, but not necessarily developing disease depending personal humoral immunity. Exampes: HPW, chlamydia, etc. More? Ask doc timely.
It is possible. It is advisable not to share drinks or food,
Nope, no worries. They call them "sexually transmitted" for a reason: they are passed from person to person only by sex. It's possible to have oral infection with some STDs, but even when present, they are not transmitted except by sex. Even open-mouth kissing doesn't transmit STDs. Oral-genital contact sometimes does so, but not all that often. For sure the is no risk from sharing drinks.