Starting with your low voltage (24), from thermostat it sounds like you have it correct. Red wire to "R" terminal on transformer. White wire to coil on new relay. Then an added wire from new relay coil to "C" on transformer. This will energize the new relay when there is a call for heat. Jumpering the "T T" on new primary good.
Now, line voltage. you have 120 vac coming into furnace, most of the time they use a light switch for the disconnect at the furnace, does yours have this switch so you can turn on and off? From there it should go into the back side of the transformer into a 4x4 electrical box. The white wire will or should stay neutral all the way through the wiring of furnace. The black will have at least 3 more wires wire nutted with it. 1. will be for the transformer the 2nd will be for the fan limit and the 3rd will be for the burner gun. since we are not using the relay from fan center we will not deal with any other wires behind the transformer. The wire going to the fan/limit is the wire I wish to deal with. Do you know what the fan/limit is? where it is located in your furnace? and what it does? Basically, it is the control switch that turns your fan on when the furnace gets to the set temperature. And it is also your furnace "LIMIT" switch. If your blower fails or if you are not moving enough air because of a dirty filter the fan/limit will rotate to the end and power would be disrupted and burner will turn off. There are two sides to the fan/limit, the left side of dial is you fan side. Bottom wire should be line voltage coming in and right above that should be a wire that goes to you blower motor, marked "fan".
To the right of the dial is the "limit" side. I hope the internal jumper is still intact so there isn't any wire on the bottom. Only the one wire on top in the marked "limit" slot. Is this the case? If there is a wire connected on the bottom right please, if you can, check the voltage and see if it is line voltage or low voltage. If line voltage we are still good. Now may I ask, did the person you bought the primary and relay from want you to control the furnace by opening the limit circuit or the main power to gun circuit? If you open the limit circuit to turn the furnace off you should always get a limit code at the new primary. if you turn power off to gun then it just turns off. I am going to stop here because I would like your feed back on the questions I have asked. But please keep in mind, The new primary does not have a limit in it, other than the cad cell circuit. The wire, red if I remember correctly, is to go to an external limit. So it is important that you answer my questions. I will be online as long as possible this morning but will be back on later this afternoon. servtech50