The fluid is brown because you never wiped it on a white rag before. It's been browning. Brown is because there is a lot of clutch material in the fluid. This happens each time the clutches apply/release. Very normal.

Because this trans hasn't been serviced normally and you are having trouble, a pan drop or flush could cause slippage. The new fluid will have a lot of detergents and lubricants that have long been used up in your current fluid. A flush does not kill a trans. Waiting until you have an issue to perform normal maintenance is what kills it. aka you have a problem and now are looking for magic. It may work, may not. Pan drop is the safer way to go as it leaves about half the fluid in the trans.

Here is what I like to have for the service.
2 gallons of fluid. Dex 3 or 6. You choose.
1 large Steralite or rubber maid pan. Found in the storage section of Walmart for about $10 or less. It works great, don't judge it based on it's original/expected use.
10mm deep socket
2 cans of brake cleaner
Rags
Ratchet
Torque wrench capable of 120 inch lbs/10 foot lbs. Because there are 12 inches in each foot.
Ramps or jack and stands.

Get the car in the air, put your catch pan under and bring a few rags. Safety goggles are a good idea as well. Remove the bolts starting at the passenger end and work your way about 1/2-3/4 to the driverside on the back end of the trans. Then move to the front and get abotu 1/2-3/4. At this point likely the fluid is beginning to drip. That is good. Keep removing bolts. The pan will not drop until you have the last 2-3 loosened. By that time most of the fluid will have drained. Let it drain so that when you take those last few out the rest of the way..you don't have much fluid left.

Lower the pan and dump into the catch pan. Lay it on a large rag to be pulled out from underneath the car. Notice if the gasket has ridges in it. If it does, likely it will feel like it is rubber over a layer of metal. This is the factory gasket and expected to be lifetime. Wipe it off and look it over. Save and reuse if it's not trashed. It is a better gasket than the one in the service kit with filter.

By now the fluid is done draining. Hop under and wiggle the filter out. The piece Scotty talked about should stay in the transmission. It is the sealing collar to hold the filter in the trans. The rule and most directions in the trans service kit will tell you. "If the new filter fits into the old collar snugly, leave the old collar in the trans. If the filter is loose..put the new collar in".. If you need the new collar, say the word and I'll post some helpful tidbits on that. 95% of the time you don't change it.

Let's clean that pan. Wipe the pan off/out to get the excess off it first. Take the magnet out and clean it down. Brake cleaner is fantastic for removing oil/dirt from things. Make your pan shine inside and out. Put the magnet back into the pan and sit the pan down where you can reach it from under the car.

Spray some brake cleaner on a rag and wipe down your tools. Put all your bolts (20 x 10mm) into that rag and spray them off with brake cleaner, then roll them around a bit. This cleans the bolts and tools. Makes reassembling clean and easy.

Ensure the filter is in place and snuggish. Put your gasket back on the pan. Wipe down the gasket surface on the transmission. Bring the pan up and start one bolt at each end to hold the pan up. Now start all your bolts and lightly snug them. Then torque them to 10ft lbs which is 120 inch lbs.

Crawl out and bring your pan and tools with you. Put your funnel into the transmission dipstick tube and fill with approximately 6 quarts of fluid (1.5 gallons). Lower the car or start it and back it off the ramps. Now check the fluid and top it off to the top line of the full mark. Ensure later that the trans is filled properly, because adding fluid into the same tube that you measure the level with can make it difficult to read.

You are done. Clean up.