My advice is "fake it 'til you make it!" Why do you feel like it must be "extremely awkward" for your female patients? I don't think it is. Especially for just a quick 5-min assessment. The patients are used to being asked questions and having their vitals/assessments done. Be friendly and act confident, even if you don't feel it on the inside. Maybe you could practice your assessments on family/friends? I assess my siblings all the time to practice. Eventually, you'll get so used to assessing people that the confidence will come naturally and you won't think about the fact that your patient is a woman (which shouldn't matter anyway).
About injections- Unless your nurse specifically says that she'll do the injection, I recommend you do it anyway. Maybe she was watching you a bit closer since she knew it was your first one. Other than accidentally sticking yourself there's a really low chance of "messing up" so just go for it when you get the chance. Needles go in super easily through the skin and they're super quick. Another opportunity will come for you to try again, so don't worry!
Oh yeah, I also look a lot younger than I am and I'm 21. I found that standing up tall and wearing makeup makes me seem older. Since you're a guy that last part might not help you too much though lol. Just remind yourself that you worked hard to get into nursing school
, you earned your spot, and you're paying to be at clinical so you can learn. That helps me feel more confident.