I'm sorry to hear that you are concerned about some of your new pup's behaviors.
Did you abruptly change her diet when she came to live with you or is she eating the same food she was used to?
Is the panting behavior all of the time, or is it only when she is sleeping, or after activity?
Is she coughing?
Any history of heart disease or a heart murmur?
Has she been tested for heartworms this spring?
Panting can be a sign of anxiety, pain, nauseousness, fever or low oxygen levels.
Please check her gum color. They should be a nice bubblegum pink color. If pressed on they should whiten and then quickly return to pink (less than 2 seconds).
If her gums are pink and blanche well then her oxygen levels are likely fine and she has something else going on.
If they are gray or white then possible heart disease, internal bleeding, red cell destruction or lack of production would be possible and she should be checked by a veterinarian promptly.
Is she eating and drinking normally? If so that is a good sign. Dogs that are in real trouble health-wise usually lose their appetite.
Since she vomited and seems lethargic, or if she seems to be drinking lots of water, or has a tense, painful abdomen with gentle pressure then there is likely an internal problem going on that needs to be investigated further with a veterinary visit.
But if she had a quick change in food and she's only had the one vomiting episode then she may just have mild stomach upset and there are some things we can try.
To try and settle her stomach today you can give either:
1) Pepcid-ac (famotidine) at a dose of one half of a 10mg tablet per 10 to 20 pounds of body weight every 12 hours.
2) Prilosec (omeprazole) at a dose of one quarter of a 20mg tablet per 10 to 20 pounds of body weight every 24 hours.
Either of these will reduce stomach acid and should help if this is related to simple nausea and gastrointestinal irritation.
Then start a bland diet of 1/3 boiled, lean hamburger (or boiled, white skinless chicken), all fats and juices drained off, mixed with 2/3 boiled, plain white rice. Feed several small meals a day. If she does well with this then slowly transition her back to her regular diet. This was likely related to nonspecific stomach upset, which should be investigated further with her veterinarian if it returns.
Can you take her temperature? A normal rectal temperature in a dog is 100F to 102F, anything at or above 103F is considered a fever and indicates infection or systemic inflammation.
If she doesn't have a fever and she is eating then ok is she having trouble jumping, going up or down stairs or rising from a lying down position? If so and all else seems normal then musculoskeletal disease, possibly a back pain or an injury are likely behind her symptoms. If you can find a particular area that seems painful warm compresses or a heating pad set on low on that area for 10 minutes at a time several times today is indicated. Shih tzus as a breed are prone to back problems. You should greatly limit her activity (no running, jumping or stairs) and keep her quiet for at least 7-10 days.
If this has been going on for several days a hands on veterinary examination is the best thing. I understand that it's evening now and that may be difficult today so I've given you some ideas to try and pin point a little further what may be causing her distress and the urgency of each possible underlying problem.
I would recommend having her seen tomorrow though if she's not eating well, continues to vomit or has poor color.
Please reply with further questions or specific responses to the questions I've asked above for further guidance.