The part "pushing my speaking voice instead of accessing your singing voice" What exactly do you mean with that? Personally I considered either lowering it an octave or going into head voice but neither seemed to sound any better. Could you try clarifying that a bit more?
"Breath support" is the basic singing technique whereby you produce high air pressure, but controlled, low airflow. That's just part of the training, and there are countless online articles and videos about it (often conflicting and dependent on genre).
The higher tones you wish to produce, the more important "breath support" becomes. This is because you need high air pressure in order to generate higher frequency sounds.
The problem is that high pressure normally wants to produce high airflow, a combination that is not good for the soft tissue of your vocal cords. Your vocal cords may reflex open to release the pressure, and you will flip into falsetto. Alternatively, you can learn "breath support" to control and slow down the airflow. How you do that is very genre dependent. Some emphasize diaphragmatic/abdominal control, others vocal cord musculature and cord closure.
If you avoid high pressure altogether, you will find yourself "pushing" for those high notes. It will become particularly apparent when you have runs of notes that vary quickly, or particular vowels (yours seem to be the closed ee and oo vowels), or if you have an aspirant onset (where a word begins with h - what little pressure you are using will escape through the h). Actually, you sometimes use an aspirant onset even with no h -- just a vowel. This is another type of "pushing" that replaces breath support. Some examples...
0:52 civilization - quick run of closed vowel, intensity drops
0:55 "east at least" - timing goes out and there is a wobble because of an unsupported, closed ee vowel
2:25 "hear" - the h steals away the pressure and the following syllables wobble
"Breath support" is basically about having a "reservoir of pressure". If there is not enough support, you will sometimes sound strained or as if you are running out of air.
But always count your blessing. It looks as if you have a good enough ear, and you timing is not going to be an issue. Maybe playing the guitar has helped there.