Hi Matheson, welcome to Permies!
Carbonation happens when enough CO2 gets suspended in a liquid, and then the CO2 will come out of suspension forming small bubbles on nucleation sites. Have you ever poured a carbonated drink, beer or soda, into a glass and then have it totally foam over, but it wasn't foaming over in the bottle? That happens from an abundance of nucleation sites in the glass allowing for lots of CO2 to come out of suspension immediately, from either scratches in the smooth glass, dust particles, or even from
soap residue from washing and rinsing, like those
water spots sometimes seen in glasses. This is how carbonation happens.
On to your brew that seems low in carbonation, and that is resulting from low amounts of CO2 in the liquid. The CO2 in there making carbonation comes from the microbes (yeasts) fermenting sugars. Without enough residual sugars when bottling, there will be poor or no carbonation. Too much residual sugars when bottling can result in exploding bottles, and I've had cases of exploding beer bottles back in my brewing days.
My impression is that I need to wait longer until bottling. Does that sound right?
This will have the opposite effect. Allowing it to sit longer before bottling will allow the yeasts to munch through more sugars, resulting in flat or still beverages. Sometimes still beverages are nice for the style, like some wines. But it sounds like you're after carbonation and mouthfeel. There are a couple ways to approach this. Bottle sooner, which means leaving more sugars still floating around that haven't been consumed by the yeasts, and that will make for more fizz. Another way is to allow fermentation to fully finish, then add sugars right at bottling, and this can allow for a measured amount of sugar giving a known amount of carbonation. Then you can tweak future batches, adding a little more or little less to achieve the level of carbonation you desire. Hope this helps!!