Yes, you could apply for asylum based on changed country conditions. Consult a qualified immigration attorney.
As for your TPS, when did you apply? It should get granted assuming you were otherwise eligible at the time.
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Start by talking to an experienced immigration attorney locally.
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You can apply for asylum only when you are in US. Certainly you may claim changed conditions in your country. A competent immigration lawyer is a must. You still have the burden of proof.
You ought to qualify for TPS and it can take up to 6 months. Have you been fingerprinted yet? On asylum, one has to apply within one (1) year of coming to the US. One of the few exceptions is that there are "changed country conditions" i.e. things have changed and deteriorated in your native country of Syria since you left. That's a given but you also need to be able to show how the civil war and strive is not just generalized to your entire country BUT you specifically, i.e. so you "may" qualify if you, personally are afraid of returning to Syria for specific reasons based on 5 specific grounds, e.g. political asylum, social group, religious, etc BUT not simply because things are bad in Syria. You should definitely consult with competent immigration counsel. At least you still will probably qualify for TPS so nothing to loose if you apply for asylum BUT don't rush into it AND clearly have your particular situation and facts analyzed by competent counsel BEFORE you make a hasty decision.