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Will the transactions I sent with my old wallet go through once my upgraded wallet is synced?
Likely not since if you indeed generated a High S tx then you will need to perform another tx
where your signature uses the v0.11.1 mandated low S.
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If I click Abandon transaction on the payment I attempted to make earlier - would the coins then be returned to my wallet for me to spend again?
Again, the transaction has not hit the block chain yet, let alone received a confirmation.
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Yes, Abandon transaction and create a new one if the coins did not "appear" in your wallet.
They will not be accepted on the network until you redo with a new tx, according to HI-TEC99.