I suspect that the linkages are not hooked up correctly. This manual should help you double check your assemblies. https://mymowerparts.com/pdf/Kohler-Service-and-Repair-Manuals/Kohler-Service-Repair-Manual-Command-CV17-CV18-CV20-CV22-CV25-CV22-CV23-CV26.pdf
Without the engine numbers I’m assuming you have a Briggs engine with the primer bulb mounted in the air filter base. Most times the base will warp where it mounts to the carb, allowing the primer to leak. Two ways I’ve fixed this problem. Instead of one gasket between the base and carb body...
Simple test before throwing time and money at this engine. Before opening the engine hook a VOLT meer across the battery and watch the voltage when trying to start the engine. If the battery voltage drops below 10 VDC when the engine stops turning, your battery needs to be replaced.
Please post the model, type and code number from the engine. Also post the part number for the new coil you purchased. It would not be the first time I’ve seen DIY guys get the wrong parts. With the numbers we can check which coil you need.
If used properly, I’ve found that leaf blowers definitely have a place in lawn maintenance, especially as you get older. Raking at my age, mid70’s, is too hard on my body and the blower makes it much easier to clean up my lawn.
I would agree with Star, bad battery. An internal shift solenoid needs a minimum of 11VDC to kick in and hold the bendix gear in place. Other cause would be that the internal solenoid contacts are corroded, which mean taking the starter apart.
For setting the diaphragm needle on two cycle and some four cycle diaphragm carbs. New style on page 31 of this manual. https://www.miniplane-usa.com/pdf/WalbroServiceManual.pdf
Or google search this, 2 cycle carburetor needle adjustment tool