What should I watch out for regarding swallowing issues?

You should set up an appointment if a doctor to scope her throat. To make sure there are no other diseases causing this coughing episode.
As we get older we lose lubricant for our throats a lot of times medication has a lot to do with this. Your best bet is to see your PCP and address this issue.

Does your mom have Alzheimer's or another type of dementia? I ask this because loss of the swallow reflex is a possible effect of this type of disease. The safest thing to do is to have her evaluated by an SLT (speech, language, therapist). This can be done at her bedside. If she can travel, then a swallow test can be performed at local hospital. Inform her PCP of your new findings. When she's eating, no one should be talking to her as this will distract her from the task at hand. Conditions such as acid reflux (GERD - gastroesophageal reflux disease) could also cause the coughing and often is treatable with medications such as a proton pump inhibitor (omeprazole, prilosec, dexilant). The biggest concern with her coughing and change in swallowing habits is the risk of aspiration pneumonia - where the food accidentally get into her lungs. This is usually the start of a downward spiral leading to a decision of whether or not to place a feeding tube... But before going down that road, I'd bring her to her PCP for an evaluation. If she develops a fever with her cough, this is a bad sign and she needs immediate attention for possible aspiration pneumonia. Start with her PCP and take it from there. Best of luck to you!

You might want to be careful about her body position when eating or drinking. If she is in bed or a recliner that is more likely to cause problems than in an upright position as if at a table and chair. That is advice straight from my speech therapist last week.

It is generally the liquids that will cause choking.
The body begins to forget to detect the difference between thin liquids and air thus it does not close the passage between the throat and the lungs.
Thickening liquids will help. thickening will go from what is called Nectar thick to Pudding thick.
You might want to discuss this with her doctor most likely a swallow test might be ordered but it is not necessary you can just begin to thicken liquids.
Jello is considered a thin liquid as it will begin to dissolve to a liquid. Same with Ice cream. Also stay away from slippery things like peaches, nectarines and plums just to name a few
Not thickening liquids and the subsequent choking can lead to Aspiration and Aspiration Pneumonia.

Agree with AlvaDeer.

Pasa, have you had anyone listen to her chest in the last few weeks? It could be a swallowing issue but I really wouldn't make any assumptions.

You need to make a doctor appointment immediately. The doctor will order swallow evaluation tests. This is often done by Occupational Therapy. It may be done with or without xray. Until you can have this done your mother should not have anything but thickened liquids given slowly and carefully. No straw. If this causes swallow problems you really are in an emergent situation and need to consider ER; call her doctor first.
End of life swallow problems are not at all unusual and some dementias such as Lewy's has often a swallow problem component. But this is a huge danger to a senior. This coughing and choking is allowing small amounts of food to be not swallowed into the stomach, but to enter the lungs. Pneumonia is the result, and it can kill. It is called Aspiration Pneumonia. You can go to google or other search engine online to look this up.
Good luck. Do not ignore this. I hope you will update us.

Our speech pathologist and speech therapist (through home care) told us that crackers, toast and other foods with little crumbs can cause coughing and "frank" aspiration.

If you're wondering about pureed foods, and she's coughing, it's time to see a speech pathologist, ASAP, or a doctor who can recommend one and have a swallow study done.   It's better not to wait, for any reason.   

If she does have some dysphagia, a speech pathologist can diagnose that and advise what level she's at, as well as the proper diet, i.e., whether mechanical soft, or pureed, and pureed to what level.

If she does like crackers, try putting some applesauce on them; it will absorb the crumbs.   My father used milk and I think some other liquid for his cereals, but our speech therapist advised these dry foods needed to either be avoided or the crumbs encased in something like applesauce.

When she was weak and bedbound, was she suffering from pneumonia?

It is generally the liquids that will cause choking.
The body begins to forget to detect the difference between thin liquids and air thus it does not close the passage between the throat and the lungs.
Thickening liquids will help. thickening will go from what is called Nectar thick to Pudding thick.
You might want to discuss this with her doctor most likely a swallow test might be ordered but it is not necessary you can just begin to thicken liquids.
Jello is considered a thin liquid as it will begin to dissolve to a liquid. Same with Ice cream. Also stay away from slippery things like peaches, nectarines and plums just to name a few
Not thickening liquids and the subsequent choking can lead to Aspiration and Aspiration Pneumonia.