I know that they had full legislative power during the republic

No they did not. The senate was an advisory body composed of former magistrates who held no legislative power at all. Magistrates above the rank of quaestor, once their year in office had ended, sat in the senate, where they issued advise to serving magistrates, usually in the form of consulta. Consulta had no legal force and even the senatus consultum ultimum had no actual legal power behind it--these are not leges, though few magistrates would dare to submit a bill to the comitia to become a lex without a consultum from the senate supporting the motion. Laws were proposed by the magistrates and passed by popular vote in the various comitia. Though the senate had immmense influence they had no legal right to enact legislation, and technically their opinions were just that, opinions--by Caesar's time magistrates began simply ignoring senatorial consulta and bringing laws directly to the comitia, unafraid of the consequences of disobeying the senatorial body

It's hard to say "during the Empire," because of how long the Empire was around. Generally, it just constantly lost power, sort of eventually forming more of a body of dignitaries and advisors than an actual legislative body.

At least during the Byzantine times, it's rule was one of slowly losing power, although it was still prestigious. Justinian's Corpus, for example, gave the Senate the right to debate all laws before they were instituted by the Emperor (although this ended up not happening), and judicial trials were often held by a prefect and 5 senators drawn by lot. High treason trials were often done in front of the Senate as well, meaning it had sway in legal matters like that.^1 It's real power, more than anything though, came from the people who made up the Senate moreso than the prestige of the body itself. Senator was a title that was, like many Byzantine courtly titles, given out like candy at times to members of the government, so the Senate often acted like the conglomeration of the government.^2 As such, it's few moments of relevancy in the Byzantine times was during times of succession. For example, the Senate was the body that decided that the gates would be open to Heraclius during his war with Phokas. Likewise, at the death of Heraclius, the intervention of the Senate is what forced Empress Martina to accept the rule of Constantine III and later Constans II over her own son Heroklonas.^3 You see it pop up from time to time after this, such as when Theodora (Empress of Theophilos) went to the Senate to appeal to their virtue in hopes that they would allow her to peacefully rule as regent for her young son Michael III without fear of intervention/scheming against her. Following this, however, the Basilika law code that was introduced under Basil I and Leo the Wise stripped it of almost all of its remaining power. After this, you could simply buy the title of Senator, so any real power and or function was more of a product of the people who bought the title than it is the title itself.^4

Hope this answers PART of your question :)

[1] The Civil Law, Justinian et.al

[2] Alexiad, Anna Komnenos, along with The Byzantine Aristocracy: IX to XIII Centuries by Michael Angold (I've only read excerpts from this, but this seems like a good book for you to get your answer from in terms of later Byzantine history)

[3] History of the Byzantine State, George Ostrogorsky (This isn't a great text since it's outdated, but it serves its purpose as a survey piece)

[4] A Synopsis of Byzantine History, John Skylitzes with translation by John Wortley