Bug: Account SDK returns incorrect or missing public address after reconnecting signer

### Describe the bug

When an account with a persistent signer (e.g., injected provider or local key store) is used, and the application is reloaded or reconnected, the Account SDK sometimes returns an undefined, empty, or stale public address instead of the expected current address. This can cause downstream operations (such as preparing transactions or displaying the user’s wallet address) to fail or show incorrect information.

### Steps

1. Integrate the Base Account SDK into a test application and configure a persistent signer (e.g., MetaMask or Base Wallet).
2. Connect the wallet and retrieve the public address from the Account SDK — it displays correctly.
3. Refresh the page or programmatically reinitialize the SDK with the same signer.
4. Attempt to read the public address again via the SDK’s API.
5. Observe that the returned address is empty, undefined, or differs from the expected current address.

### Expected behavior

When reconnecting a signer or reinitializing the SDK with a signer, the Account SDK should consistently return the correct, current public address associated with that signer. There should be no stale, empty, or undefined values, and applications should be able to reliably fetch the address after every reload or reinitialization.

### Version

_No response_

### Additional info

_No response_

### Desktop

_No response_

### Smartphone

_No response_

Start by reproducing the reconnect and reinitialization sequence in a test application with a persistent signer such as MetaMask or Base Wallet. Trace the Account SDK API used to read the public address after reconnecting, and compare it with the signer’s current address. Done means the API consistently returns the correct non-empty address after reloads and reinitialization.

Written by the indexing model from the issue text.