Saying that you captured most of Tenn., and have the Confederates bottled in Richmond, I'd say you are doing better than the north was in history (At that point in 1863, they were on the verge of losing the war). Keeping long supply lines is the burden of the attacker, something that halted Grant's campaign in his 2nd Vicksburg attempt and Sherman was stopped by this for a while in Atlanta. Don't be discoureged if you cannot get into a decisive-war winning battle with the south. The American Civil war was one of the only wars in history where an army never suffered more than 40% in a battle (unless it was a siege such as Vicksburg, but I was referring to the field). You probably will never get a battle where you inflict more than those sort of casualties. Only once did I ever nearly destroy an enemy army (183,000 losses out of 212,000 enemy army) and I played Forge of Freedom campaigns 9 times. As for the supply issue, you can combat this in two different ways-
1. Continue what you are doing. You will probably win the war in the end by continuing your process, but it will take a while.
2. Ignore supplies and live off the land. Risky, but that is what made Napoleon's Grandee Armee survive for so many years and have freedom of maneuver. The only downside is that if you get to an area with poor foraging, your army can disintegrate. But much must be risked in war, however, so in the end the decision is yours.
Whatever you end up doing, remember this: Despite the fact that 'Battlefield' generals like Napoleon and Lee who try to win the war on the field are some of the most skilled and most awe inspiring, in the American civil war, it's not the field, it's the factories. Keep your army supplied and destroy your enemies supplies, and you will have victory. Cut their supplies with a whole army, force them to attack you at whatever the cost because they need the supplies and factories to survive. You must give incentive to your enemy to attack you. Good luck.