What you can do is try and hold a steady contact with her mouth and squeeze with your calf. If you are posting at the trot when you rise up lean more forward, but keep contact on her mouth and squeeze at all times.
I dont usually do this with someones suggestions but I am going to do so in this case
Please do not do what has been suggested in the above quote.
Without knowing a bit more about your mare I can only give a general suggestion going on the assumtion the horse has had little training.
One of the first things a horse (not just an ottb) needs to learn is
Forward doesnt mean to slow nor does it mean to race around.
I am really bad at explaining things so I hope that someone else will better put this explanation into the proper test for reading and comprehension
If your horsse is heavy in your hands I am guessing that you might not have the softest of hands and or your horse is not giving with her jaw.
Does she have her head up in the air looking around?
If you were to give the horse a continual squeeze and lean forward your not going to teach it anything other than how to ignor your constent squeeze
This is not the way to give and aid to move off your leg! If you continually squeeze it is almost as if your teaching the horse to block out the "white noise" whcih can equal "dead to your leg"
The leg aid should first ask quietly, and then escalate in its intensity!
Take into account that you must give a tad bit of time to have a reaction from the horse before giving the stronger aid.
I will ask with a slightly stronger intensity and then after the third time I might use a whip and give a tap either on the horse or on my boot or behind the saddle pad....just depends on the horse. The same intensity scale is used with a whip. Do NOT just all and out smack the horse the first time out. Although this is an appropriate response to some horses for certain things and certain training issues. Without seeing or knowing the horse I am going on the assumtion it has no formal re-training from the track.
I do not lean forward while asking. leaning forward might get you iin a bad position for you to fall off.
As for slowing and speeding up a trot I would first suggest to get the horse to be able to hold a nice steady rhythm that works for her to be able to learn preferably a bit more active than slow.....not a western jog.
Also when you talk about "extended" this is something this horse is not capable of at this point. She most likely does not have a clue what you mean from the way you describe her.
Getting her to step actively from behind and under herself might be obtainable but not extension.
Now for the speeding up and slowing of the trot rhythm. Now this is something she may already do on her own while trying to figure out her rhythm, but it is different when your asking for that rhythm to change when you want it too.
There is the aspect of slowing your posting and speeding it up but it is oh so much more than that. There is the use of your abdominal muscles and the zipping of those abs....
When you would like to slow the trot you will engage the abs and post smaller/slower and to produce a larger trot using the abs and seat a slightly larger (and slighly I mean it does not mean post to the sky) post asking for a bigger trot. I say bigger instead of "faster" because these two things are differnet. Bigger does not equal faster.
If you take up a steady contact with this already heavy in the hand horse be sure that you are giving and relaeasing because if you simply grab up contact and hold it it will likely have the opposite result. A horse who is being carried by you and your hands...
If your trotting along you can try this method to soften the horses jaw.
Hold a steady but not overkill contact on the outside rein. With the inside rein you would like the horse to give to your rein aid. I will use the word "tug" but I do not mean sharply. It should be light enough that it is not a shock but strong enough to feel. A jiggle of a a tug so-to-speak.
With the rhythm of the trot. Holding light steady contact on the out side and a rhythmic tug - tug - light .....tug-tug-light
Say the words out loud as your trotting in a nice rytmic cadenced trot.
The "light" part is when you give with your rein aid there.
You must learn to feel when the horse gives you even the slightest of a try to relax the jaw and at that moment you need to release and give them that bit of a reward.
If your not good at feeling it you may need a well versed eye on the ground to help you.
Please if you do not have a trainer get one. Even if you can only afford one lesson a month that is better than NO direction at all.
I am not a trainer nor a teacher. There are many on this forum who are judges etc etc.......
These are tolls that have taught to me by instructors and trainers over the years.
Again I am not very good at explaining things so I hope I have done well enough for you to understand.
Try to establish a nice rhythm of the trot with your mare.....dont complicate it at this point by trying to get her to change the rhythm to a more energetic one or a slower one. Just work on a steady even cadenced rhythm for now.
All the best....