I live in Oregon, and our senators are Jeff Merkley and Ron Wyden. I understand that they have offices in Washington, D.C. If that's the case, who are the senators and representatives working in the state legislature here in Oregon? I don't suppose they have offices both here in Oregon and D.C.
TL;DR: In the state capitol.
State senators and state representatives have nothing to do with the federal government. They are part of the state government. They can only pass state level laws.
In the United States, the state governments are separate entities from the federal government. Each state has its own state constitution (although they may not call it that). They have their own executive, called a governor (although there is no federal requirement that they use that title, all states currently do). Their own judges. And their own legislature.
This contrasts with how many European countries are organized. In Europe, the subdivisions of a country are often administrative parts of the country as a whole. This may be a result of feudalism, where the lords derived their power from the monarch. By contrast, in this US, the states united to form a government, ceding some of their authority to the federal government and keeping the rest.
It happens that Oregon chooses to use the names Senators and Representatives for their legislative members. These are the same as the federal names. But there is no official relationship between the members of the state legislature and the federal legislature. They may choose to work together, but there is no legal requirement that they do so.
The federal senators and representatives likely have offices in Oregon as well as Washington, DC. The Oregon offices would be used for constituent outreach. While constituents can travel to Washington, DC to talk to their senators and representatives, most prefer to communicate with them in their home state. The DC offices are used more for coordinating with other members of the federal Congress.
The state representatives and state senators may also have local and state capitol offices. The local offices for constituent outreach in their district and the capitol offices for coordinating with other state legislators. State legislatures are often less funded than the federal Congress though. So they may work from home in the district and share offices in the capitol.
I'm going to assume that by "State Senator" you mean people like Jeff Merkerly, who represents Oregon in the US Senate. (This is an incorrect use of terminology, but otherwise you question doesn't make sense.)
A Senator will normally have multiple offices.
For example Jeff Merkerly has offices in DC, Portland, Eugene, Medford, Salem, Bend and Pendleton.
Naturally when the Senate is sitting the senator will spend most of his or her time in Washington DC, however the Senate schedules time for state work throughout the year. At these times the senator could travel back to their home state to consult with local people.
The State Senators and Representatives are a completely different set of people, elected separately every two years (elections are held in November). In Oregon they work part-time. They can pass any state law that doesn't conflict with national law or the constitution.
They in fact have multiple offices. https://www.wyden.senate.gov/contact/office-locations and https://www.merkley.senate.gov/connect/office-locations
They do this to connect with their constituents (and donors). They need permanent staff (and therefore offices) to deal with the daily flood of locals trying to get a moment of their senator's time or give him a piece of their mind.
Oregon isn't a particularly populous state, or a national politics strategic state so it might be expected they need few offices, but it is a large and geographically diverse state and since they represent the whole it is valuable to have local outreach almost everywhere.
People in the Oregon legislature generally just maintain one in Salem (the state capital).