I'm a tenor with a range of A2-D5(F5) I can almost belt an A4. The songs I perform require me to sing with a full tone through all of the phrases, even into the high notes. A weak falsetto won't cut it. What exercise would help me get a full tone on every note? And I've heard about octave sirens, but i can't tell a difference after a week.
What exercise would help me get a full tone on every note?
Vocal exercises specific to full tone would mostly include laryngeal position and resonance. I would define full tone as a healthy, resonant, non strained quality. There are a variety of online resources specific to a lower laryngeal position as well as proper abduction and adduction in vocal folds for clarity.
A brilliant professor once had me go through my entire range starting from low to high and sing each note for 8 counts going from pp to a comfortable forte. We went up chromatically with the goal of having a clear sound as resonance was my goal. Once you can master this you will be closer to a full, healthy tone on any note within your range.
"Weak falsetto" is a mischaracterization: I exchanged my self-accompanying instrument from fingerpicking guitar to accordion when changing from tenor to countertenor exactly because the reduced dynamic range of the falsetto was typically louder.
However, falsetto most certainly has a sound significantly different from a belted chest voice. If you are stating "require me to sing with a full tone [...] even into the high notes" you are operating your voice in a manner that has a lot of potential to be damaging.
You only have one voice, so it does not make sense to get your advice from strangers over the Internet for operating your voice at maximum strain levels. Avail yourself of a vocal coach.
If you are not singing opera, it might make more sense to work with your voice further within your comfort zone. And if you are singing opera, you have no business foregoing vocal coaching.