but the water flows into the tank just past the "settle rinse" label on the dial.
It would be helpful if you post all of the labels stated on the dial. A "Settle Rinse" (aka: Rapid Rinse) setting would be also utilized for a filter control so by posting all of the settings, should confirm if you were supplied with the controller for a softener and not for a filter.
The usual cycle settings (and common cycle times) for a 5600 softener are:
SERVICE (Normal operation when supplying soft water to the house)
BACKWASH (10 minutes)
BRINE DRAW (60 minutes)
SETTLE RINSE (10 minutes)
BRINE FILL (time varies depending on the BLFC installed and the salt setting programmed)
after BF, the controller will return to SERVICE mode until the next regeneration cycle is scheduled.
The common tank for 1.5 cuft of resin is 10" X 54" and 10" X 47" for 1.25 cuft. While you stated your tank is an uncommon 10" X 53", it most likely contains 1.5 cuft.
If your brine tank has an overflow fitting somewhere on the side, then the safety float should be adjusted so the incoming water will be stopped before draining out that fitting. If there is no overflow fitting, the float may be as high as it will go since it is only a safety device to prevent the tank from overflowing if there should be a malfunction. If the float is adjusted too low as you are describing, it can shut off the incoming water flow below the level the fluid should reach with the proper salt setting, thereby producing insufficient brine for the amount of capacity to be regenerated.
The float normally slides on the rod which is lifted when the float hits a rubber/plastic ring placed tightly around the rod. That rubber ring can often be slid farther up the rod thereby allowing the float to rise higher before lifting the rod which closes the safety valve.
Another float style has the float permanently fastened to the rod, with the rod running through a hole in the lever which activates the safety valve. Rubber rings are placed on the rod, above and below the lever, so the ring below the lever will be raised by the float, lifting the lever to stop water from flowing into the tank. That lower ring can often be slid downwards, thereby allowing the float to rise higher before stopping water flow. The ring above the valve lever can also be lowered, to hang the float and rod on the valve lever, keeping the float elevated above the usual fluid level in the brine tank. Since the float will lift the rod, excess rod length above the safety valve will sometimes need to be trimmed shorter to prevent the rod from colliding with the brine well cover thereby preventing the safety valve from activating.
We will need to know your water's actual hardness amount as well as the amount of iron and manganese if either is present, so as to calculate the regeneration frequency requirements.