What I am needing to know is, does my transmission have a dipstick?

This video shows the job being done on a similar car but the process is the same. The fluid fill will be explained in the instructions and diagrams below.

https://youtu.be/HJdcWI6m3J4

Here is a guide to help you step by step with instructions in the diagrams below to show you how on your car.

https://www.2carpros.com/articles/how-to-service-an-automatic-transmission

Refill

NOTICE: This procedure contains the air purge steps required to purge air from the transmission fluid cooling system. This procedure is NOT intended for use with the Transmission Fluid Level Check.

NOTICE: The vehicle should not be driven if the transmission fluid level is low as internal failure could result.

NOTICE: The transmission fluid fill plug is located near the exhaust system. The exhaust will be extremely hot during this procedure.

NOTICE: The use of any other transmission fluid than specified can result in the transmission failing to operate in a normal manner or transmission failure.

NOTE: If the transmission starts to slip, shifts slowly or shows signs of transmission fluid leaking, the transmission fluid level should be checked.

NOTE: Here is an overview of the Transmission Fluid Drain and Refill procedure.

- Adding 3.3L (3.5 qt) of transmission fluid to the transmission is an initial fill enabling the engine to be started.

- The cold level range shown in the procedure allows the vehicle to be driven.

- The vehicle should be driven to allow the TFT (Transmission Fluid Temperature) to reach 96.7°C-102°C (206°F-215°F) in order to purge the air from the transmission fluid cooling system.

- Fill the transmission fluid to the fill range on the transmission fluid level indicator at the normal operating range 96.7°C-102°C (206°F-215°F).

1. NOTE: The transmission will need 3.3L (3.5 qt) of transmission fluid added to the transmission as an initial fill if:

- a new main control assembly has been installed.
- the transmission fluid pan or transmission fluid filter have been removed.

NOTE: The transmission will need 11.35L (12 qt) of transmission fluid added to the transmission as an initial fill if the transmission has been overhauled.
Using the Transmission Fluid Fill Tube, add the transmission fluid to the transmission through the transmission fluid fill hole. Refer to Adding Additional Transmission Fluid in this procedure.

2. NOTICE: Only 9-10 ml of the 6R ATF Additive Kit XL-16 kit should be used after all of the fluid has been replaced in the transmission. Too much additive in the fluid can result in converter clutch slip issues.

NOTE: The 6R ATF Additive Kit XL-16 kit contains two syringes. The syringe with additive "F" is to be used after all of the fluid has been replaced in the transmission. The syringe with additive "T" is a tracer that is part of the kit and is not used.
If all the transmission fluid has been replaced add 9-10 ml of Motorcraft(R) 6R ATF Additive Kit XL-16 syringe with additive "F" to the transmission.

3. Check the transmission fluid level cold.
- The vehicle is safe to drive if the transmission fluid is in the cold level range 32°C-43°C (90°F-110°F).

- Using the scan tool and with the engine running, place the selector lever in each gear position and hold approximately 5 seconds. Place the selector lever in PARK, with the engine at idle (600-750 rpm).

4. Separate the transmission fluid level indicator from the transmission fluid fill plug.

Check out the diagrams (Below). Please let us know if you need anything else to get the problem fixed.

So is this the type of transmission I can only take to a dealer or can I take it to any mechanic?

No, you or any mechanic can do this service. no need for the dealer. the cost will be really high.

Roy