I don't think there are any concrete plans at all for the segment of I-369 on the North side of Texarkana. AFAIK TX DOT still doesn't have plans yet for extending I-49 up across the Red River in Arkansas.
For now it seems like TX DOT is only concentrating on a couple segments of I-369 to the South of I-30. They're still trying to figure out how to get I-369 to depart from the TX-151 loop SW down the US-59 corridor. They done studies and reached a consensus to upgrade along the existing US-59 roadway, but that will mean buying and clearing a whole lot of properties. Extending I-369 North of I-30 appears to be an even more difficult problem. It looks like any Northern quadrant of a I-369 loop would have to extend North from I-30 out miles West of town.
Bypass plans in Marshall appear father along. Construction on that could begin in the next couple years.
As I understand it, TXDOT has purchased the I-49 path north of Texarkana and is allowing the previous owners to continue using it for agricultural use until the time comes to build the road. That may be decades from now.
US-59 / Loop-390 around Marshall TXDOT looks to be planning to build from US-80 to IH-20 beginning, as you said, in the next couple of years. It initially would use a I-20 link to the existing US-59 near the south of town. The portion from IH-20 to the current US-59 probably would be farther down the line.
As to I-369 there are zero plans beyond I-30. Texarkana interests are firmly wanting it to come to LOOP-151 as 59 currently does. There is discussion for I-369 to extend slightly west of FM2148 to meet I-49 north of Texarkana Texas, but that is a non-starter until I-369 or I-X30 meets Loop 151.
As to extending I-369 directly north of I-30, that is not really feasible now. The original plan of I-49 (or the "NEW US-71" as it was called back then), was for it to follow the current Loop 151 from US-71 west around Texarkana Texas and then to continue to the currently identified Red River Crossing.
When Mike Huckabee was Arkansas governor, all of this changed. Huckabee had Arkansas start building I-49 around the East side of town. Any sense of cooperation between the two sides (which up until the late 90's shared a police force) soon dried up. Huckabee even had the ZIP codes changed in Miller County from "Texas based" 755XX numbers to "Arkansas based" 718XX ones. It got even worse when Former Texarkana (Texas) ISD Superintendent became the Texas side city manager. Today, The fire departments do not aid one another. Mutual aid from Nash or Wake Village goes to TA but the two Texarkanas do not provide mutual aid for fires. Arkansas side