I've had both knees replaced. Before that, I had injections, I think something like synvisk(???), it has been a number of years. I too was afraid the shots would hurt. They didn't. It seems like I had to have the shot once a week for 3-4 weeks. It did help some.
My sister-in-law had the shots more recently & it did not bother her either and helped her more than me but she is a normal weight. I would definitely give it a try. The shots are nothing like you are thinking they will be..

I had a series of injections in my knees (halogen). My knees were too far gone so the shots didn't do much for me because my arthritis was too advanced.
But for many people I know find the injections helpful and can have weeks and weeks of relief.
Way before that I got a Cortisone shot in the knee which helped at that time. Years ago.
So I would say yeah give the injection a try whatever it is the doctor is suggesting.
I got both knee replacements in 2010.
I Also had a Cortisone shot in my shoulder. Great help.

Note. Cortisone shots are only good for once in a blue moon. Not good to do many but they can be great and most helpful.

So what injection is the doctor meaning. A cortisone shot?
Or a series of injections over a few weeks to help lubricate cartilage?

If you get a doctor who knows what he/she is doing and has done plenty of knee injections then it ought not hurt. Maybe some discomfort. Only one time I went ow when doctor injected me but that was it And it didn't put me off for the next injection. So no. Not a lot of pain involved.

Good luck. I hope things go well for you.